That's very difficult question to answer. You have 26 seasons of Classic Who to choose from, along with 8 different Doctors (if you include the movie).
Each Doctor is different, and each person you ask may end up recommending a different Doctor for you to start with. For example, I grew up watching the latter days of the fourth Doctor (Tom Baker), but I remember watching more of the fifth (Peter Davidson).
Some of Tom Bakers latter stories can be quite dark and gothic, but nothing a 10 year old shouldn't be able to handle.
A good story to show your sister would be 'Remembrance of the Daleks' (a seventh Doctor story), in which was the last time you saw Davros in the classic series, and shows a small war between both Dalek factions. It is also where, due to the Doctors actions, the main trigger for the Time War occured.
Have a look at the BBC site for the classic series:
~Link~If you can't access that, try the Doctor Who Wiki:
~Link~