A great 'on foot' sightseeing tour: Walk from Trafalgar Square through St James' park down The Mall (take some bread and feed the ducks) to Buckingham palace. Then walk back along Birdcage Walk, past the guards buildings (check about changing times) to the Houses of Parliament/Big Ben; then walk up the Embankment a bit and back(don't bother with Downing St it's boring); then cross the bridge to the South Bank - there are sometimes worthwhile things on there IIRC and on to the London Eye. South of the river is less interesting so don't bother walking around there.

I've not toured Buckingham Palace but I have been around Windsor Palace - truly amazing. If you can tour it, do so.

Battersea Power station is cool if you like your architecture/photos.

Denise suggested Eltham Palace if you are into Art Deco; we've not been but it's on the list. It's a tube/train ride away.

Museums in close proximity:
* Natural History
* Science Museum
* V&A (Victoria & Albert)
Nearby is the Royal Albert Hall.

Places to shop:
* Covent Garden
* Carnaby Street is interesting
* Harrods used to be good
* Bond St is nice to window shop
* Avoid Oxford/Regent St. Same old big dept stores.
* look down the alleys - it's amazing what you find.

Eating reasonably close:
Soho is good for restaurants - not all expensive.
Garfunkels are reasonable
Texas Lone Star in Kensington is fun