There are really four epocs of Doctor Who.
Epoc 1: The Black & White years. Doctors 1 & 2 (Hartnell & Troughton). The show in its earliest days. While there are things to be enjoyed here for hard core fans (like me) it's probably not for everyone. Unfortunately, many of the shows in this era do not survive, so it's impossible to fully appreciate or judge the first two doctors as fully as we might.
Epoc 2: The Cheesy years. Doctors 3 & 4 (Pertwee & Tom Baker). Don't get me wrong, I enjoyed this era immensely. Jon Pertwee is probably my favorite Doctor. But the budgets and production values of this era is like watching a B-movie when you're used to big-budget Hollywood blockbusters. It can be painful. Still, a lot of seminal Doctor Who lore stems from this era, and for a true appreciation of Who, one must really start at the beginning of Pertwee's era and follow through to the end. (If you start with Tom Baker, you miss such formative events as the debut of The Autons, The Master, Sarah Jane Smith, and much of the background of The Doctor himself, as well as the Time Lords.)
Epoc 3: The 80's. Doctors 5, 6 & 7 (Davison, Colin Baker, McCoy). Peter Davison was fantastic. Colin Baker was a joke. Sylester McCoy was intriguing, unfortunately the show ended (at least the original run) on his watch, so we didn't see as much of him as we should have. Cheesiness is still evident, but the show was evolving in this era. If you like the Modern Who enough to want to explore more, you could do well to start with the first 5th Doctor episode and see how far your interest takes you from there.
Epoc 4: The Modern Era. Doctors 9, 10, 11 (Eccelson, Tennant, Smith). If you can't enjoy this show from the 2005 reboot, then you have no reason to watch anything earlier. The show has evolved into the modern era and become far more than anything before it. As others have suggested, you can watch from 2005 and get the vast majority of good "Who" without missing much.
I have been a Who fan for over 20 years, and watched every extant episode at least once (as well as reconstructions of most of the "lost" episodes). I won't encourage anyone to take their interest in Who as deeply as I do (I mean, get a life, already). That being said, my vote would be to give the 3rd, 4th, 5th and 7th Doctors a chance. They all have their charms, and if you like what you see, the others might have something to offer you, as well.