As a fan who has watched what I can find of the 1st and 2nd Doctor episodes and almost the entirety of everything beyond that, I'd make the following recommendations. There's no right or wrong way really, but these are my two cents on possible options on places to start:
1. Check out from the 2005 reboot - Basically, if you'd rather watch the newer episodes (if you don't think you can handle the dubious effects and sometimes questionable acting) this is a good place to start.
2. Series 3 (2007)- this was the first GOOD Tennant Series. All you really missed from Series 2 was that Rose, one of the most unlikable companions in the show's 48 year history, is trapped in another dimension and the Doctor misses her for some reason.
3. Spearhead from Space - Correct me if I'm wrong, but all (or nearly all) of the episodes from this onwards are around. This is the start of the 3rd Doctor's run and if you're interested in the older Seasons, this is a pretty good place to start. Don't worry about characters like the Brigadier who you won't recognise, you'll understand them soon enough. The Autons (Nestenes), despite appearing only twice in the classic series, became an iconic villain, and for good reason
I may however be slightly biased because this is my personal favourite Doctor.
4. Genesis of the Daleks - Nobody can call themselves a true fan until they've seen this one. Arguably one of the best serials ever, and certainly one of the best to feature the daleks, if you don't like this one the classic series probably isn't for you and should maybe try suggestions 1 or 2 before giving up on the show.
This is all I can think of just now.
EDIT: Actually, since people seem to be talking about best Doctors, I don't think there is a best. It depends not only on the actor but how you like their portrayal of the character. Tennant, for example, is a fantastic actor (watch Single Father if you don't believe me), but frankly, I thought much of his run was pretty poor and he didn't really feel like the Doctor to me so much.
Personally I'd rank them thusly:
1. John Pertwee (3rd)
2. Tom Baker (4th)
3. Matt Smith (11th)
4. Peter Davison (5th)
5. Christopher Eccleston (9th)
6. Patrick Troughton (2nd)
7. David Tennant (10th)
8. Sylvester McCoy (7th)
9. Colin Baker (6th)
10. William Hartnell (1st)
(Oh noes! I didn't put Colin last :O! But seriously, listen to the Big Finish Doctor Who Audios. He's FANTASTIC in them!)