I got interested in Dr Who in a round-about way.
I never watched the original, because I lived in several areas that did not have PBS.
About a decade ago, I got interested in a SciFi series called Special Unit 2 — a special force that went after bad aliens. It was a comedy-drama.
Then I heard about Torchwood and that it, too, was good guys chasing bad aliens. I watched it. It made numerous references and sort of filled in a lot of info on Dr Who.
Thus, I started watching the ‘new’ Dr Who series with Christopher Eccleston. He was the Doctor for only about one season. Then came David Tennant and later Matt Smith.
I am hooked.
I also liked the Saturday morning children’s series, The Sarah Jane Adventures — which was also good gal chasing bad aliens. I didn’t realize at the time but Sarah Jane Smith [Elisabeth Sladen] was one of the companions in the original Doctor Who series.
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Start with the Christopher Eccleston era. Many later episodes refer back to things/aliens/cultures/events in the original and earlier ‘new’ series episodes.
I loved Special Unit 2. It's run shows up on TV about twice a year.