What are the best things to do in London? Watch
The nicer and more beautiful/iconic parts of central London to explore are Sloane Square/the King's Road, Notting Hill, Marylebone High Street, Islington, and all around The City/Square Mile (where you'll find St. Paul's Cathedral, Monument, the Bank of England, the Barbican Centre, and all the skyscrapers like the Gherkin and the NatWest Tower). For more contemporary/hipster areas where there are plenty of trendy eateries and street art, try Shoreditch, Dalston, Borough (there's a famous market here on Saturdays), and Southbank, perhaps even Clapham.
Depending on how much time you've got, it might be nice to do a tour of the Royal Parks. You could cover the central London ones in a day (Hyde Park, St. James' Park, Green Park), then spend a day each at Richmond and Greenwich Parks (they're well out of central London but there's a lot to do and explore in Richmond and especially Greenwich other than the parks, so you'll easily fill up a day at each).
If you do decide to go to Greenwich, travel there from central London on the Thames Clipper service (not one of the tour boats) which is very fast and takes in multiple landmarks along the river.
Hope you enjoy your time here
Go shopping in Oxford circus (although be prepared for the prices) or even window shopping. dining at all the restaurants London has to offer. visit some of our parks (hyde, greenwich, st james, kensington gardens), visit kew gardens. visit the famous markets like camden, brick lane, borough market in London Bridge.
go on the London eye? a boat ride across the thames - typically 'touristy' but also enjoyable
visit the theatre. if you want to visit the cinema (movie theatre) try the Lounge which has in-room dining