If you haven't yet seen all of the David Tennant episodes, it's probably best to bookmark this and come back later. As River Song would say, "Spoilers!"
In the first few episodes with David Tennant, both he and the writers seemed to be finding their legs. Right away, he brought a lighter feel to the Doctor, as someone much more adept at comedy than Eccleston had been. To be fair, however, Eccleston's Doctor was much closer to the Time War, when he'd made the terrible choice that led to the end of both the Daleks and the Time Lords. Rose Tyler helped him start to make peace with that decision, but it was a dark time to be The Doctor.
"Doctor Who," Season 2
One of the immediate changes in Season 2 was the Doctor's relationship with his companion, Rose Tyler. Whereas Eccleston played the Doctor as protective in almost a fatherly way (until the very last episodes, where a romantic vibe developed), Tennant's Doctor had a much more flirtatious relationship with Rose from the start. He was clearly jealous of her on-again-off-again boyfriend, Mickey, and while he wouldn't outright commit to any romance with her, the possibility dangled tantalizingly throughout Season 2.
My favorite Season 2 episodes were "Tooth and Claw," where they encounter Queen Victoria and what appears to be a werewolf (I dearly loved how Rose tried throughout the episode to get her to say "I am not amused"); "The Girl in the Fireplace," where they try to save Madame de Pompadour from clockwork robots (the different perceptions of time set up future episodes where similar ideas were explored); of course "The Rise of the Cyberman" and the follow-up episode, "The Age of Steel," which set up the parallel world; "The Satan Pit," an outer space adventure, in space suits, where the Doctor battles an alien who may be the origin of Earth ideas of the devil; and "Army of Ghosts" and the follow-up episode, "Doomsday," where the Doctor and Rose once more fight the Cybermen and, in order to save the Earth and protect her, the Doctor sends her and her mother into a parallel universe, where they must stay.
The best episodes of Season 2 explored big concepts, such as parallel universes and the consequences of challenging the rules of the space-time continuum. The final scene of "Doomsday," where Rose and the Doctor must sacrifice their continuing relationship for the sake of two worlds, is as big as it gets.
I also loved that year's special, "The Runaway Bride," which introduced Donna Noble and provided some much-needed levity. As I watched it, I thought that was when she became the Doctor's companion, because I had heard her name mentioned by my friends, but it turned out it was a one-off, and she wouldn't return until Season 4.
I was less fond of "Love and Monsters," told as a vlog by a young man who joins a fan group for the Doctor and ends up getting into serious trouble of the alien variety. It was an interesting concept, but there was too little of the Doctor in it, and it made the Doctor seem more flippant and unfeeling. I also disliked "Fear Her," a tale about a little girl who can trap people in pictures. It felt more like a "Twilight Zone" episode from the 1980s and never quite rose to the level of the rest of the season.
Those who have been watching the show longer than I can tell me whether fans readily accepted Tennant as the Doctor. I thought that, once he found his own approach to the Doctor, he was much more multi-dimensional in his portrayal than Eccleston had been.
"Torchwood," Season 1
By the time Gwen Cooper (played by Eve Myles) discovers the secretive Torchwood Institute, it has gone through major changes from the blaring white hallways of the bureaucratic organization which unwittingly opens a rift between parallel worlds at the end of "Dr. Who," Season 2. Now, instead of a well-funded organization staffed, presumably, by hundreds, Torchwood has literally moved underground, operating from the Cardiff docks and staffed, at the time Gwen comes along, by only three people. The head of Torchwood? Captain Jack Harkness (John Barrowman), the self-aggrandizing swashbuckling "pretty boy" met in "The Empty Child" from Season 1 of "Dr. Who."
From the start, Jack Harkness seemed more like a cardboard hero than a fully-realized character, perhaps hearkening back to classic British boys' books about daring adventurers. So while I enjoyed the first episode, where Gwen does her own investigation into the shadowy organization, the minute we began to see more of Captain Jack, the more the charm of the series started to fade. The problem is that Captain Jack works better as a secondary character, providing comic relief as back-up to The Doctor. Whether it was the writing or Barrowman's acting abilities, the attempts to provide him a deeper back story continually fell flat. So, for example, "Small Worlds," where Jack reconnects with a past love, came across as almost laughably wooden.
By contrast, the best episodes focused on other members of the team, such as "Greeks Bearing Gifts," where computer expert Toshiko acquires an alien pendant allowing her to read people's thoughts. "They Keep Killing Suzie" had some good ensemble acting, as they tried to figure out how to stop a woman who was brought back to life by alien technology.
Because the final episode of that season, "End of Days," featured Captain Jack essentially saving the world by himself, I found it far less moving than I'm sure the writers meant it to be, however.
"Doctor Who," Season 3
In episode 1 of Season 3, "Smith and Jones," we meet Martha Jones, a doctor in training who shows bravery and creative thinking, helping the Doctor return a hospital to Earth after it is transported to the Moon by aliens. While some fans probably disliked her from the start, simply because she was not Rose, I liked her bravery and quick thinking. I found it implausible, however, that she immediately developed feelings for the Doctor which, of course, he could not return, still not over Rose.
My favorite episodes from Season 3, again, were those big-concept episodes, such as "The Evolution of the Daleks," where the Doctor is faced with a choice to help the Daleks reach a new level of existence; "42," a tightly-wound episode where the Doctor and Martha try to unmask saboteurs on a spaceship headed for a fiery fate; and "Human Nature" and the follow-up, "The Family Blood," where the Doctor becomes human to hide from enemies who are tracking his DNA signature. Despite the fact that the Doctor and Martha were barely in that episode, I enjoyed "Blink," which introduces us to the Weeping Angels, causing fans everywhere to leave a light on at night.
There were episodes in Season 3 that I felt didn't work as well, such as "Gridlock," where Earth residents reside in flying cars, perpetually stuck in traffic. That one felt a little too cartoonish. I also disliked "The Lazarus Experiment," where a scheming professor thinks he has found the secret to an extended life. The aging makeup was not great, and I found Martha's family members, who were endangered in that episode, to be more annoying than sympathetic.
I'm more split about "Utopia," which features Captain Jack Harkness catching up with the Doctor while he recharges the TARDIS at the Cardiff time rift. They travel to the far future where they strive to help Professor Yana rescue the humans. Captain Jack, of course, was back to sidekick status and therefore more tolerable. Professor Yana turned out to be a very interesting character, setting up the last two episodes of that season. I also liked some aspects of "The Shakespeare Code," because of its re-imagining of how one of the Bard's plays became lost, but I disliked how flirtatious he was with Martha, since it seemed that the writers were trying too hard to prove to us that she was sexy, even though the Doctor had no interest in her.
Martha proved that her bravery and quick-thinking were extremely valuable in the last episode of the season, "Last of the Time Lords," that jumps a few years into the future to a dystopian world run by the Master. She must do the main work of saving the day for the Doctor, who is imprisoned and forced to age into a wizened, CGI version of himself.
At the end of that episode, when Martha decided to stay home for her family, who were emotionally wrecked by their experiences serving under the Master, it was definitely anticlimactic. Hers was one of the more subdued departures of a companion, from what I understand. This could be why my husband barely remembers her as a companion.
"Torchwood," Season 2
The writing for Season 2 of "Torchwood" improved, with better story lines and fewer ridiculous and/or cliched lines coming out of the team's mouths.
Some of the best episodes featured just one member of the team, such as "Last Man Standing," a moving episode where Toshiko falls for a soldier who's trapped outside of his time but agrees to help heal a time rift, despite his feelings for her. Also noteworthy was "A Day in the Death," where Owen, who's been brought back by the alien Resurrection Glove after losing his life helping the team, copes with mixed feelings about his new existence.
The arc of that season was more impactful, since it was woven throughout the whole series. Captain John Hart (James Marsters, who played Spike on "Buffy the Vampire Slayer") was an interesting nemesis, and while the final confrontation was all about Captain Jack's past, the focus on the team as a unit made the season finale, "Fragments"/"Exit Wounds," more compelling.
"Doctor Who," Season 4
For the final season of David Tennant's time as the Doctor, he teamed up with Donna Noble (Catherine Tate), also known as the Runaway Bride from the special after the end of Season 2. I was looking forward to seeing her episodes, having liked her as Nellie Bertram on the U.S. version of "The Office." Knowing she has comic chops, I expected magic from her work with Tennant, and I wasn't disappointed. Setting up terms from the start, that they were only interested in being friends, freed them both from the sort of awkwardness engendered by Martha's crush on the Doctor.
What made her even better was her ability to do more than get laughs. One moment, Donna was bluntly calling out the Doctor, bringing him down to earth. Then, another moment, she was talking to the people and aliens they encountered, forming real emotional connections. Donna was the one, for example, who tells the Doctor, in the episode "The Fires of Pompeii," "You're the Doctor. You save people." Those watching the current series will recognize that Peter Capaldi's Doctor revisited that phrase, in an episode where he goes to extraordinary measures to save a young Viking woman. "I'm the Doctor, and I save people," he declared.
My favorite episodes from Season 4 include "The Fires of Pompeii," where the Doctor and Donna land in Pompeii on the tragic historic date, only to discover that the volcano is not erupting; "Planet of the Ood," which highlights Donna's tenderness as she and the Doctor try to help free a slave race of aliens; "The Doctor's Daughter," where the Doctor's DNA is used to create a female warrior for a war that has lasted "for generations"; "The Unicorn and the Wasp," an Agatha Christie murder mystery-inspired episode, featuring Christie as a character; "Silence in the Library"/"Forest of the Dead," where we meet River Song, who would become a recurring character; and "Turn Left," where Donna finds herself in a parallel universe where she never met the Doctor, getting help from Rose to set things right.
Martha made some appearances in this season, now working for the British military defense force, UNIT, tasked with defending against aliens. Her contrast to Donna was clear: Martha was all business, while Donna got more emotionally involved with what was happening as they fought the Sontarans, resolved a conflict between warring races on a planet, and then saved the world from a new Dalek invasion.
The final episode of the series, "Journey's End," has a somewhat misleading title. The episode marks the end of Donna Noble's time as a companion, not the end of Tennant as the Doctor. For the episode, the Doctor got help from former companions Sara Jane Smith and Rose Tyler, as well as occasional companion Captain Jack Harkness. There was something both preposterous and joyous about seeing them combine their forces, along with a brave-to-the-end Harriet Jones, former prime minister. Sadly, the end of Donna's time as a companion was something she herself did not want at all. The Doctor wiped her memory of him, in order to protect her from the after-effects of her regeneration of his cut-off hand into a Time Lord-human hybrid. This process also gave her access to the Doctor's memories, which were far too much for her human mind to contain. If her memory stayed in tact, her brain would become overwhelmed, and she would die.
The Christmas special for that year, "The Next Doctor," featured David Morrissey as a man who claims to be a Time Lord called The Doctor. The episode started out lighter, with its references to classic Doctor Who, and then, as the companionless Doctor unraveled the mystery, became very moving.
Then came a special called "Planet of the Dead," where the Doctor goes on a bus trip and winds up on a strange desert planet. With help from a burglar, Lady Christina de Souza, he has to figure out how to defeat a hoard of metal-eating insects and get the passengers home safe. Lady Christina was like a female Captain Jack, and I'm glad this was a one-off for her character.
A special that aired in November 2009, "The Waters of Mars," has to be considered one of the best David Tennant episodes. In it, he finds himself on Mars at a crucial time in the history of Earth's space exploration. He must choose whether to try to help the ill-fated astronaut/scientists or allow history to take its course. This is the episode where the Doctor decides that, as the only remaining Time Lord, he can give himself permission to break the rules of time, saying, "Time answers to me, and I will win!"
This also helps to set up "The End of Time" parts I and II, where the Doctor teams up with Donna's grandfather, who unlike previous companions' family members, had always admired the Doctor. Facing off with the Master, he soon discovers they have bigger problems: the return of the Time Lords, as well as Gallifrey itself, threatening to destroy the Earth. I initially watched this episode out of order, since it was shown as one of the curated "Best of Dr. Who" episodes shown before the start of the current season. Back then, I did not recognize the various people that the Doctor says good-bye to once he knows that he will soon regenerate. Having seen all of the Tennant episodes (the tenth doctor), however, I found that part of the episode bittersweet.
"Torchwood," Season 3 (Children of Earth)
Season 3 of "Torchwood" was more like a miniseries: five episodes all linked, dealing with one major threat. A mysterious alien race speaks to Earth via its children, sending an ambassador to negotiate a terrible bargain: 10 percent of the world's children or the extermination of the human race.
The Torchwood team, now down to three -- Gwen, Ianto Jones and Captain Jack -- face off with government entities, trying to kill them, as they strive to uncover the truth about the aliens and their demand. In the course of the miniseries, Torchwood is once more destroyed, and the team loses another member, not to mention Captain Jack being faced with a gut-wrenching "Sophie's Choice" type decision.
After seeing this miniseries, it's amazing the writers managed to find a way to pull off a Season 4. The Earth may have been saved, but Torchwood was essentially destroyed, both physically and emotionally.
Incidentally, this miniseries came between the "Dr. Who" specials "Planet of the Dead" and "The Waters of Mars," which helps explain why the Doctor finds Captain Jack drowning his sorrows in an intergalactic when he comes to say good-bye.
I loved Tennant as the Doctor for the reasons listed above: his ability to blend lighthearted moments with seriousness; to be at the same time very kind and occasionally fearsome. His run as the Doctor, probably more than any of the recent Doctors, sparked a resurgence of interest in the show. It's hard not to admire the Tenth Doctor, who was faced with some of the most terrifying threats and yet, through a combination of cleverness, humor and loyalty, kept fighting and, for the most part, winning.
On the other hand, if I weren't a completist, I would probably have stopped watching "Torchwood" after Season 1. That would have been a mistake, as the writing got better in Seasons 2 and 3. However, I'm not surprised the series ended after Season 4. It's hard to build a series around a sidekick, especially when you're constantly making allusions to the original hero, the series can only pale by comparison.