The problem is that the BBC wiped a lot of the early stuff made in the 60s, because of the price of video tape. A lot of really good stuff was lost, not only Doctor Who, but some really good comedy as well. So if you wanted to watch from the beginning you couldn't.
A good place to start would be with the Christopher Ecclestone series as Doctor Who. He's the first modern doctor, the show was cancelled for a long time, and like Star Trek the Next Generation, it's quite easy to pick up even if you never watched the original series. If you want to, you can then dip in and out of the older episodes. The doctors are as follows
Patrick Troughton (my own personal favourite, but it's probably got as much to do with me being a kid when they came out, as anything else)
These three all have gaps, and it's worse with Hartnell than Troughton, and worse with Troughton than Pertwee.
Peter Davison (I stopped watching at this point, I was at University)
Show then cancelled for a long time, apart from a one off millennium special starring Paul McGann.
Matt Smith (the current doctor)
Peter Cushing also played the doctor in the two Hammer movies.