Thirded on Borough Market. I used to work around there, and have to say it's one of the finest places to go for good fresh produce. If you can face the queues, try and buy some cooked food while you're there. Much better than anything you'll get in a lot of the adjacent eateries.
For historical sightseeing, I would suggest cross referencing
Peter Ackroyd's London with a decent street map. This might take a while though, so I'd further cross-reference it with a Time Out/Rough Guide...
Get a copy of Time Out (weekly) when you're here and trust it's art/film/music sections. I think it's an indispensable guide to what's on.
All the recommendations here are pretty good, but I thought I'd add my personal (touristy) favourites:
The ICA (on The Mall just down the street from Buck palace) - the programme here is always varied and rewarding. It's an arts centre that's become a bit commercial of late, but it still manages to impress me with its roster of films, talks and exhibitions.
The National Film Theatre (in Waterloo) - Go for a (GOOD) film, but stay for a drink by the river and a casual perusal of the second-hand books being sold directly outside.
Edgware Road (at the northern end of Oxford St) for some nice Arab cuisine and a toke on a shisha. Maroush is great for food (but it's always packed to the hilt, so go in the afternoon)