I find your position easy to relate to. I've spent years being aware of the show, somewhat tempted to try it, but the number of season has always put me off. I've spent a large chunk of my life watching all the Stargate (among other series), so I knew that such number of seasons is a big commitment, particularly when I find it hard to relate to the older stuff.
But then when Season 5 started and all my sci-fi-watching friends were jumping around from happiness, I couldn't resist anymore. So I started at the rebooted Season 1. In retrospect that definitely was a good place to start. It's fairly new, so I could relate to it rather easily. Everything is explained as if it's the first time you see it, all characters and concepts are well introduced. Plus the series picks up the pace quite fast and as soon as you reach Season 2 and the Tenth Doctor (David Tennant) you will find out why he's one of the most popular doctors. Once I went through all the new seasons, ending with the end of Season 5, I suddenly had a lot of time to spare before Season 6 came along. And so I started exploring the original series, the first eight doctors.
Someone recommended to me to check out the Fourth Doctor and particularly to start with the episode the Ark of Truth and then watch the next few episodes. That was a great idea. I was introduced to the character in an interesting, but not too important episode, then suddenly I was watching one of the key origin episodes in which the Doctor is present at the creation of Daleks. After watching the series since reboot, this was really a big deal. Afterwards I watched a few episodes featuring other doctors. Eight Doctor is just a movie, which is easy to find. The others were a lot harder to track down.
It turned out that BBC has taped over a lot of their archive tapes, destroying many of the key episodes. That became the number one reason why I didn't try watching the whole series from the very beginning - a lot of it was missing. Amazon surprisingly features a decent collection of a few episodes in its videos on demand. I got a gift card for it when I was buying a textbook, so I tried it and in the end was quite pleased, as I found some episodes I couldn't find anywhere else.
And so watching the rebooted series and then picking up a few (often random) episodes from the original series worked out quite well. The new series provided a good foundation, so I could follow all of the old episodes...