Disclaimer: I don’t live in London, and I don’t really know what’s cool. Apologies if any of the following is obvious or inappropriate.
That said, you won’t go far wrong knowing about these record shops:
Honest Jon’s is a must. They have two locations - one on Portobello Road, which also has a cool market, the other in Kings Cross.
Reckless Records on Berwick Street (Soho)
Sister Ray also on Berwick Street
Rough Trade has two locations, one on Portobello Road, the other just off Brick Lane.
Music & Video Exchange / Soul & Dance Exchange, Notting Hill.
There’ll be loads of cool stuff that I don’t know about around Dalston, Hackney and Shoreditch.
Books: Charing Cross Road has plenty of used book stores.
Museums: I love the V&A, and I absolutely love Dennis Severs’ House, but your mileage may vary.
Eating: don’t know, this all changes so fast that you have to be a local to give tips, really. But you can’t go wrong with a trip to Brick Lane. Bottom end you’ve got a load of Indian restuarants and Indian supermarkets. Middle bit you’ve got 93 Feet East, Rough Trade East etc, plus a load more Indian restaurants. Top end you’ve got two fantastic 24-hour bagel bakeries, all sorts of cool bars and independent vintage / art / design / clothing shops, the Boxpark and Flashback Records.
Clubs / art spaces: no idea, although I hear E1 and Printworks are good.