You can always use the Netflix streaming option to filter your choices. The newest four seasons are currently available via streaming on Netflix, so I'd start there.
My wife got me interested in this show and that's where I started. I have to admit, when she makes me watch the old ones, I do often find them cheesy. If you just start with the old ones, I think you won't appreciate them enough to watch them all. However, if you start with the new ones, they will introduce you to most of the classic villains. Once you know them and enjoy them, you'll be interested in understanding their back story.
Take for instance the Daleks. Their orignal story is really good, but I enjoyed it as much as I did because I knew them from the newer seasons. I just don't think that the story lines enough will be sufficient to keep your interest given the old special effects. But add in there the desire to understand their origins, and you have motivation to help through the cheesy parts.
Just my $.02.