I recommend first of all getting an Oyster card, I think it costs £5 but the travelling will then be a lot cheaper for you.
There's quite a few good things to do in London, Museums have already been mentioned. The War Museum is great, it is in Elephant and Castle though, and that's a pretty dang rough area, so be careful!
The London Dungeons is quite a fun place, it's expensive though. There is a place similar to that which is called The London Bridge Experience which I think is a little less pricey.
Travel to Camden for a day and evening, it's a pretty interesting place with strange, weird and wonderful shops and people. London Zoo is also kind of near there.
Covernt Garden is a very nice place to spend an evening. Pricey though, so check out prices before you go!
Try to avoid places like Croydon, Tooting, and pretty much all of the south, it's filled with chavs and gangs down that way. Avoid Croydon at night.
The Shakespeare Globe Theatre is a wonderful place to go, for the history and also the tickets to see plays are fairly cheap, you can get some for about £5- £15.
Take a walk along the Southbank, it's beautiful and there is always something interesting going on. Near there I think there are some gardens which are free to go into.
Richmond Park is a gorgeous place to go, it is in a crappy area though, but Richmond Park is lovely and it is massive.
There are a lot of good pubs and things around Leicester Square, which is also a good place to go and from there you can walk to pretty much anywhere in London.