My parents (bless them) named me after a Doctor Who character, but although I really liked Eccleston I never really got into the show until Tennant's last episodes. Then I decided to watch them all from the beginning.
Almost five years later, I'm still only up to Tom Baker. (What? It's a lot of episodes!) My deepest respect to those who've also made it through all of the audios; I really want to do that too but assembling an achievable list is difficult. (I'd like to hit all of the audios where a Doctor reprises his role, but even with that limitation there are just so many.)
First Doctor
My Doctor. A lot of people can't get past Hartnell's hostility in the first episode, but really, Hartnell played
every likable characteristic of the Doctor during his tenure.
Unearthly Child (On the strength of the first episode alone.)
Time Meddler
The Myth Makers
Second Doctor
Troughton was one of the few that didn't impress me in his first serials. Thank goodness I stuck with him, because I got to like him a lot, particularly his deep friendship with Jamie and Zoe. (Also the names of two of my siblings. Seriously, Mom, Dad, what is the deal?)
The modern Who in particular rarely seems content to let the companions just be friends with the Doctor, so it's nice to see here.
Web of Fear
The Mind Robber
The War Games (I could not stop watching this one until I got to the end. I can't imagine the surprise and suspense during the original broadcast.)
Third Doctor
I liked Pertwee right away. Loved all of the attempts at action and stunts (achieved with varying degrees of success) and the rest of the cast was the best lineup before or since. The Brigadier, Jo Grant, Benton, the Master, Sarah Jane... Amazing. (Also, the Third and Second Doctors make the best Doctor pairing. I didn't want Troughton to leave after The Three Doctors!)
The Silurians
Carnival of Monsters
The Time Warrior
Fourth Doctor
I really didn't see what the big deal was about Tom Baker at first. It took awhile, but oh man, am I hooked now. People say that he's the definitive Doctor, as though everyone before him had never got the character quite right.
They're selling Baker short, though, because all of the previous Doctors knocked it out of the park. If Baker is better than them, it's just because he's that damn good.
(Still haven't finished Baker's run. I see that several other people have listed stories I haven't seen yet as favorites--good to know there's something to look forward to!)
Masque of Mandragora
The Face of Evil
Robots of Death
I've seen two of the Fifth Doctor serials during 50th events. (
Earthshock and
Resurrection of the Daleks). Davison just didn't appeal to me. Still more than willing to give him a shot.
I also caught Colin Baker's
Attack of the Cybermen, and I liked him right away. I love seeing the Doctor's arrogance running rampant. I can see where he might be off-putting, specially when he was the standing Doctor. But as one out of twelve/thirteen, he's a great change of pace.
I've only seen
The Curse of Fenric from the Seventh Doctor...wasn't crazy about the story, but I'm looking forward to more Sylvester McCoy.
I think
Night of the Doctor deserves a special mention, even though the Eighth Doctor wasn't even in more than two live action "episodes". I would put it in the top twelve best Doctor Who episodes, full stop. It doesn't stand on its own the way other stories do, but never has so much series mythology been used so well in such a short period of time.
(Again, kudos to everyone who was able to list the audios.)
Ninth Doctor
The Ninth Doctor was my first chance to actually see an episode of Doctor Who after hearing about it my entire life. I was just starting to get into it when he regenerated and they lost me.
Rose
Dalek
Father's Day (The first episode I actually liked; I didn't really "get" the show back then.)
Tenth Doctor
This was when Doctor Who suddenly went from something I'd have to explain to everyone I met to something they knew and cared more about than I did. I've seen plenty of Tennant episodes now, but still haven't given him the attention everyone feels he deserves.

I do quite like him, though.
(Let me know if there are better episodes than these.)
Blink
The Fires of Pompeii
The End of Time
Eleventh Doctor
I liked Matt Smith from the get-go. He seems to be the youngest and oldest of the new Doctors all at the same time, and that never fails to win me over.
Vincent and the Doctor
Asylum of the Daleks
The Day of the Doctor
Twelfth Doctor
Obviously it's far too soon to call this one. However, I have the feeling that when all is said and done,
Robot of Sherwood will make the list. (That's another one that I would easily place in the top twelve.)
I would also recommend the two Cushing movies (dated though they may be, they're rather impressive compared to the television show at the time) and many of the special features on the Classic DVDs.
There's a two-part documentary in the form of a story set on future Earth, a "sequel" of sorts to The Green Death, a quest for one of the most obscure Classic Who writers, and a nice feature on Who videotape trading in the early days of the format. Great stuff.
So...what's the most affordable way to start in on these audio dramas?
TC