Doctor Who explores a lot of different moral problems, though a lot of them may be hidden by the fun, action-packed adventure laid on top of them. What fascinates me the most in the show is the concept of the Doctor as a simple traveler, exploring the universe and happening to get involved in these strange situations where he ends up saving the day. He's this strange mix of powerful Time Lord and ordinary guy wandering around, looking for new things to see and new people to meet. (It just occurred to me that he reminds me a lot of one of my idols, Richard Feynman, who would go to unusual places and talk to everyone just to experience them.) I think this is one of the reasons that the Eleventh Doctor didn't appeal to me as much as his predecessors: his stories (both arcs and individual episodes) were very directed, and his personality was based on his awareness of his superiority. He had puzzles he was trying to figure out (Amy, River, Clara) and reasons to go where he did. I did like him, but not as much as I liked the Ninth or Tenth Doctor, or especially the Eighth Doctor (if I can refer back to the classics for a moment).
I really do love the background music in Doctor Who. I'm especially enamored of the way music was subtly woven into the fabric of the Tenth Doctor's life (I could write an essay on this), but from a general standpoint, Murray Gold has done a fantastic job of using the music to enhance what's going on on-screen. (Though, "I Am the Doctor" was a little over-used.) My favorite in-show piece is "The Doctor Forever": I love how the two disparate sections - the ballad and the action part - are variations on the same theme, and they both illustrate the Tenth Doctor very well. It's wonderful the way that the Eleventh Doctor's quirky personality is reflected in the seven-beats-per-measure cadence of "I Am the Doctor". Another favorite piece is "All the Strange Strange Creatures", and "The New Doctor" makes me dance whenever I hear it (earning me strange looks from my coworkers).
Of course, the Doctor Who main theme is simply classic. I don't know what it is about it, but it's simple yet completely captures the feel of the show. My favorite version of the theme from the modern show is the fully orchestral score from Series 2 and 3 - the bridge makes my heart soar every time - but the Eleventh Doctor's and Twelfth Doctor's variations capture each of them beautifully, the first having a rock-n-roll feel and the second being a bit of throwback to the classic versions of the theme.