Well my favourite episodes are Logopolis (4th Dr), The End of Time (10th Dr), The Two Doctors (6th & 2nd Drs), The 1996 TV Movie (8th Dr), and The Tomb of they Cybermen (2nd Dr), but Most people tend to have different tastes to me. What I'd do is try one story of each doctor and see which ones you prefer. Here's a list of my recommendations:
1st Doctor - The Daleks: Most people prefer the Dalek Invasion of Earth, but this first encounter with them has such a better storyline. In regon 2, it's in a boxset with An Unearthly Child and The Edge of Destruction (the 1st and 3rd serials ever), so that's a good place to start.
2nd Doctor - The Tomb of the Cybermen: the definitive Cyberman story. A lot of people also like "The Invasion" with the Cybermen in, but that's like an part serial and drags on like forever. This one is only four parts.
3rd Doctor - The Three Doctors and The Green Death - The Three Doctors s a fun story which brings back the second doctor for a serial and also briefly the first doctor. The reason the first doctor was not in this one much is because he was ill at the time. I think William Hartnel died soon after. The Green Death because in my eyes, it's the best non multi 3rd doctor serial. People always recommend the Silurian's' and The Sea Devils, but they're some of those awful typical third doctor stories that try to deal with current political problems while also trying to be a sci-fi. Also, the Planet of the Daleks is moderately popular, but is really just a dull remake of the Daleks serial (see above) which dragged on.
4th Doctor - Genesis of the Daleks: Just because it's the definitive fourth doctor serial and quite possibly the definitive dalek serial. It can drag a little, but I've not seen it in years. Also try The Keeper of Traken and Logopolis. Two great stories part of a trilogy with two fourth doctor serials and one fifth doctor serial. The fifth doctor one is a bit silly though, but the fourth doctor serials in the trilogy are great.
Fifth Doctor - Earthshock: Duh! The ultimate fifth doctor serial in my opinion, and made one of the most hated companions ever into a hero. People also like the Five Doctors, but I think it's the weakest of all the multi doctor serials (not counting time crash). It's okay, but the other two are outstanding. Another popular one is the Caves of Andorsi or whatever, but I confess I've not seen it. Been looking out for it, but it's too expensive. £20 for a DVD, NO THANK YOU!
Not even many special features. Waiting until the special edition come out.
Sixth Doctor - The Two Doctors and Revelation of the Daleks: Despite being one of the most unpopular doctors of his time and even being the only doctor to ever be sacked from the role, Colin Baker stared in some amazing serials. The Two Doctors is a fun romp which returns the second doctor onscreen for the last time. Also some great Sontaran action in this one. Wish we saw more of them in the classic series. Only four serials, shocking! Revelation of the Daleks is also fantastic and in my opinion on of the best dalek stories of all time. Probably the darkest story of the classic series as well. Great fun to watch. Vengeance on Varos is also popular, give that one a go too, can't go wrong with it, and if you have the stomach for it, try the epic fourteen part serial "A Trial of a Timelord". Cool story ark with many sub stories in.
Seventh Doctor - Remembrance of the Daleks and Survival: Remembrance of the Daleks as it's the most popular seventh doctor serial and the last appearance of the daleks in the classic series, and survival because it's the last classic serial ever, and is just a good romp.
Eighth Doctor - The TV Movie: Well it is the only eighth doctor story on screen anyway. It is a gooden though despite what anyone says. Give it a chance.
Ninth Doctor - All of series 1: I can't name a bad ninth doctor episode at all. Just watch everything! He only appeared in one series. The best ones though are probably Aliens of London/World War Three where the Slitheen invade earth, Dalek which deals with one rogue dalek in an underground base in America which is probably the best dalek story ever, Bad Wolf/The Parting of Ways which is the Ninth doctors swansong and another great dalek story, probably as good as Dalek, Fathers Day which is a weird one, but really enjoyable, and The Empty Child/The Doctor Dances which is Steven Moffats writing at it's very best. Hugely enjoyable.
The Tenth Doctor - So many, I can't even list them all!: Okay here we go: Rise of the Cybermen/Age of Steel, among the very best Cybermen stories ever, Army of Ghosts/Doomsday which is an epic Dalek Cyberman crossover, The Lazarus Experiment which is just a great Doctor Who story and in my opinion the most chilling monster in the shows 50 year history, Utopia/The Sound of Drums/The Last of the Timelords which is an epic three parter featuring the return of a very special villain (
), The Sontaran Stratagem/ The Poisen Sky which is a fantastic Sontaran invasion, The Stolen Earth/Journeys End which has the return of all the doctors companions since the revival series, the Daleks, and their creator Davros, The Next Doctor which is a quirky story with Cybermen and Maybe another Doctor (
), and the End of Time which is Tennants swan song and the return of the Master!
The Eleventh Doctor - The Impossible Astronaut/Day of the Moon, The Angels Take Manhattan, and Cold War: TIA/DOTM is a great two parter with great enemies, probably the best Eleventh Doctor story, TATM is a great Weeping Angel story which pretty much matches TIA/DOTM, and Cold War which is the awesome return of the Ice Warriors. Reminded me of "Dalek" (see above) with the fact there is only one ice warrior on a base.