Ooh, what a cool topic! I can't believe I missed this when you first posted it...
I grew up with Doctor Who as one of those things that was always sort of around in the background. Unfortunately, the show was being cancelled just as I was getting old enough to watch TV back in the 80s, so I never really had the experience growing up of waiting for Doctor Who to be on and following the adventures from week to week. However, my dad liked it and had some on video, and there were repeats and things, so sort of by accident I became aware enough of the show to at least know which Doctor was which, and most of the companions, and to even have a couple of favourites.
Although I was familiar with the show and liked it, I never counted myself as a fan because I didn't feel I'd seen enough of it. I wanted to know more but I wasn't sure where to start.
When the news started to come through that the series was being revived, that gave me the push I needed. I decided that I didn't want to watch the new series without at least a good grounding in the old series, and so I started to track down copies of the older stories, starting with my favourite, Sarah Jane, and then moving on to Leela, Romana, Ace, and eventually almost everyone else. The only eras I'm still not very familiar with are the First and Second Doctors, and I'm trying hard to catch up!
'Life is the most wonderful fairy tale of all!' - Hans Christian Andersen, from What The Whole Family Said