Go over to Kensington.
There are several good museums there. E.g., the Victoria and Albert, Natural History and Science museums.
I agree. It's very much worth going to the museums. London has some of the finest museums in the world, and they are free admission (voluntary donations accepted).
The British Musuem near Russell Square has a superb antiquities collection - probably the finest Egyptian and Greek collections on the planet. If you have an interest in history, this is an absolute must see.
There are 3 major museums in South Kensington
The Victoria and Albert museum: an excellent British history lesson. (Another must see).
The science museum, also worth a visit (can be missed if you don't have time).
The natural history museum. (Pro-tip: use the side entrance on Exhibition road, skip if you have more important things to see)
Churchill War Rooms - The old underground WW2 government bunker in Westminster (Entry fee payable)
Off the beaten track museums:
Something a bit different, and usually distinctly British. (Well worth visiting if you're in the areas)
Sir John Soane Museum - Lincoln's Inn Fields near Holborn. (Not only a beautiful building, formerly an architect's house, but an excellent collection of historical architecutre exhibits)
Charles Dickens Museum - Dickens' old house in Holborn, with a large collection of artefacts.
London Canal Museum - near Kings Cross
London Transport Museum - near Covent Garden - see the history the transit systems of London, including the history of the world's first subway
The Wellcome Museum - near Euston. A rather eclectic collection of medical/art exhibits. Quite unique and interesting.
Shopping (not really my area of expertise)
Oxford Street - All the big brands have their flagship stores here. If you like brands, you'll love this.
Harrods - in Knightsbridge.
Fortnum and Mason - in Piccadilly. (Food supplier extraordinaire)
Markets (weekends only)
Camden Lock - off Camden High Street. Very trendy/hippy stuff. If the weather is nice, then you can take a very pleasant walk West along the canal, past London Zoo and Regents Park, and up to St John's Wood.
Portobello Road - Vibrant street market on Fridays and Weekends, but a load of peculiar and antique shops open the rest of the week. (Warning: tourists are easily separated from their money here).