I live about 40mins outside of London, so I go there a lot. Definitely stay away from touristy areas, especially during summer, or you run the risk of being trampled under foot! Places I personally like:
Camden market - it's right on the lock, so it's really beautiful round there, but also very alternative, if you're into that sort of thing - plenty of opportunity to see full-on English punks.
Covent Garden and Carnaby Street - they can get busy, but they're worth a look, even if just to experience it.
The Comedy Store - this is in Picadilly Circus, kinda down a side road. Thursday nights are good as they tend to have a lot of new blood, so it's something new to hear. Also quite reasonable, about £15 per person, but you get a good 3hours of comedy.
Also, as someone who used to commute, try to avoid buses, tubes etc unless really necessary. You can walk around London really easily (pack comfy footwear if you plan to do this). Let's put it this way, Tower of London to Baker Street, via Covent Garden, St Pauls, Trafalgar Square and Leicester Square (not in that order) took me an hour and a half with horrific blisters on my feet!
Wow, I've rambled! Hope it helps though!