The V&A would probably be the museum to fit both your interests (
www.vam.ac.uk).
My favorite museums are the Wallace Collection and Sir John Soane’s Museum.
The Wallace Collection (
www.wallacecollection.org) is a historic house and gallery with a very varied collection: period rooms, paintings, objects, there’s something for everyone.
If you’re looking for quirky, try the Sir John Soane’s Museum (
http://www.soane.org/). It’s another historic house filled to the rafters with odd, one-of-a-kind stuff collected from around the world. Plus there’s a exhibit of contemporary works in glass on at the moment
I like the Tate (Tate Britain and Tate Modern) in general but at the Tate Modern the building is the star and it’s worth a visit just for the setting but the collections are so-so. As Cheesebiscuit notes, the crowds can be annoying and it's best to go early as possible. I usually go only to see specific things. If you like Gauguin, there’s a very good exhibit on at the moment.
The Cabinet War Museum is supposed to be really interesting but haven't made a visit yet.
Have fun!