I have seen that this show is popular but upon watching some of it yesterday it seems more like a kids show that you needed to watch when you were younger or something. Definitely not what I was expecting. I had only seen the art people make for it before ever seeing anything about the show. Art is what made me curious about it. Just felt super disappointed to see what it was. Guess I should try the older part of it.
The good starting points are either 2005 (Series 1, Episode 1 - Rose) which is
very cheesy and has aged very poorly now (but it has its charm) or 2010 (Series 5, Episode 1 - The Eleventh Hour), which is more modern. A new ideal jumping on point will be coming next year - every few years the show 'resets' to a degree, though it is one massive continuity, unbroken, going back to 1963.
Ultimately, Doctor Who isn't really a kids show or an adults show; it's a family show. The best comparison I can make is to the best sort of work Pixar does - where kids love the swashbuckling heroics and the silly bits, but its full of nods and jokes that only adults will understand. Doctor Who actually does like to dabble in higher-end science fiction concepts here and there, too.
If you want a good example of the more adult end of the show, look up 'Blink' (Series 3, Episode 10) or 'Midnight' (Series 4, Episode 10). The show varies, and it's a different thing every week - so yes, you get slapstick kid-friendly stuff, but also stuff that terrifies kids and gives them nightmares, and also deeper emotional stuff for adults. It's a weird show that sort of defies proper categorization.
One thing that's worth saying is that it's more science fantasy (IE - Harry Potter with a science fiction bent, where wand-waving magic is mixed with true science and history, or like a lot of Marvel Comics) than it is hard science fiction ala Star Trek.