Dude I would really suggest staying somewhere central, Heathrow isn't very central and cen be a mission to get to and from by train espically if the tube goes tits up and it often does. One of mates lives out that way its at least 1 hour in and out.
It depends on what you wanna see really, and what kind of things you want to shop for as to where you go. I haven't lived here long, some of city guys will now the best places....I wouldn't shop in harrods any way, been turned away from there casue I wasn't dressed right!
+1 for travel card, you don't oyster card to visit, if its w/end it will be cheap rate all the time anyway.
TFL web site
http://journeyplanner.tfl.gov.uk/use...T2?language=en
I have been to go some musems,, with some of the expos, haven't around to it yet....Oh Camdam is fun if that your kind of scene! If your down on a Friday night drop by a say hi to the Soho Massive!
Well if there is anything else, you want know just ask I'll try to help....Enjoy the city!