Hi - check out the Enjoy England website, for ideas on where to go and what to do around London:
http://www.enjoyengland.com/ideas/city-b…
The British Museum, National Gallery, Tate Modern, Science Museum, and V&A museum, are all amazing places to visit, and entry is free. Do some sightseeing around the South Bank area, which faces the Houses of Parliament and is close to Covent Garden - good for theatre shows, shopping, bars and restaurants:
http://www.enjoyengland.com/destinations…
The usual things to do are actually quite enjoyable. Try The London Eye, London Zoo, the Royal Parks, Tower Bridge, Tower of London, Changing of the Guard at Buckingham Palace, shopping on Oxford Street, London markets, Greenwich:
http://search.enjoyengland.com/en-GB/Sea…
http://www.streetsensation.co.uk/markets…
As for places to hang out in the evenings, the following should keep you entertained:
Porterhouse, Jewel Bar, Los Locos and Roadhouse are always lively. Also try Belushi's for cheap drinks and the Absolut Icebar - where you're given thermal capes to keep you warm! Have dinner on an evening river cruiser, while you float along Thames:
http://www.visitlondon.com/attractions/p…
http://www.visitlondon.com/travel/gettin…
The Tube doesn't run all night - the service usually stops at around midnight, but you can use licensed taxis or night buses:
http://www.visitlondon.com/travel/gettin…
Read what other users of Enjoy England have written about their best day out in London:
http://www.enjoyengland.com/your-england…
Have loads of fun : )