Like PG said, trains and the tube will get you where you want. Renting a car /driving a car there can be a challenge. What do you have in mind for the obvious?
If I remember correctly, in Picadilly Circus there's a place that sells discounted "last minute" tickets and cancellations. While we were there, (not all obtained from that ticket place) we got 1st row seats to Phantom of the Opera (very good), saw the Mouse Trap (not so ggod IMO), and Miss Saigon (good, but not as good as Phantom)
The most entertaining IMO was Shakespeare unabridged, which is like 3 Stoodges meets Shakespeare. Very Slapstick humor and casual dress. Even 1 of the 3 guys did the part of Juliette.
I'm not sure where this is but it's in England somewhere:
http://www.carsofthestars.com/There's a nice flea market to see lots of different stuff called something like the Portabella road/flea market. Found an old 1968 Match Box car there like the one I used to have when I was little, with the box even.
Visit the local pubs for fish and chips/beer. Like BIGS said, Guiness. I don't drink beer but I hear it's different over there. Has to do with something like letting rats swim in the vats of beer, so I'm told....
There's some sort of Horror Wax museum, can't quite remember the name but I almost got my wallet pick-pocketed there!