Ditto to the above answer. In addition, you should go see Buckingham Palace, Parliament, Westminster Abbey, St. Paul's, the Globe, the Tower of London and the Tower Bridge (I've heard you can even go up to the top of the bridge). You should also see Piccadilly Circus and Trafalgar Square. I'd also recommend going to some of the parks, like Hyde Park and Regent's Park. If you're into shopping, don't forget to stop at Harrod's.

For theatre, check out this website: http://www.officiallondontheatre.co.uk. As for museums, there's Madam Tussaud's, Ripley's Believe It or Not, the Imperial War Museum, the British Museum, plus tons more. If you like art, there's the National Gallery, National Portrait Gallery, and the Tate Modern.

P.S. Before you leave, you should look at the Transport for London website (http://www.tfl.gov.uk) to learn about the public transportation system. There, you will find maps, Oyster travelcards (which you should get when you get there), and even a journey planner which provide directions on how to get from one place to another.

covent garden is great, theres a great market and cute little shops ooo also oxford street has great shopping!
umm the eye is good and allthe way along the river side :)
leicester square and soho are reli fun places
:)

i recommend camden town, such a funky place! XD
great atmosphere, shops, and hilarious people

other shopping areas include yup you guessed it, oxford street and covent garden