Since it's not mentioned here and was one of my favorite weekend activities when I lived there, I have to insist that you check out both the Camden Market and the Portobello Road Market. These are both on the fringe of the central city and are two of the best urban street markets that I've ever been to. Portobello is more antique-y and quirky and Camden has more of that counter-culture feel to it. Regardless, they're both huge tourist attractions, but they're also attractions for locals, so there's something for everyone there...including delicious food!
Other than those, I'd recommend Neal's Yard, Marylebone High Street, St. Christopher's Place, and all of the many beautiful parks that the city has to offer.