I Live in the London area...
Loads of stuff to do, and other people on this board will probably give you better answers than me as I really just live like a normal guy there, but I'll give you some practical advice for when you're actually here with how to deal with the locals - Make sure you don't appear too touristy (Fanny packs, shorts, wearing caps backward, talking stupidly loud, we hate loud, etc.) and if you're American, pretend you're Canadian. Make sure you know what area you're in at all times, London isn't like some cities with clearly defined "good" and "bad" areas, you can walk down the road and not even know you've gone into a dodgy area. Ask the locals if you get lost, most of us are polite enough (And if we're sarastic, don't take offense, it's how we are) to give you directions and even advice on what to do.
Overall, have fun, branch out, talk to people. Don't -just- do the obvious tourist stuff, see the different communities of London, everything from Middle-Class white people to Jamaicans and Irish immigrants.
Hope I helped, enjoy yourself.
EDIT: Unless you live in a constantly hot area, you won't find London weather as bad as people make out, it's a lot worse in other areas of Britain, yes I'm looking at you Scotland.