London is fantastic for art and museum fans, not least because many have free admission [though temporary exhibitions usually have a charge]. Here are a few of my favourites, but there are plenty.
The Victoria and Albert MuseumA great and quite unique
museum. You just can't go wrong here. A huge variety of things to see, wonderfully presented, not too busy and free admission. Even the museum shop is good. The icing on the cake comes on a sunny day as there's a large courtyard where you can take a rest and a cup of tea.
Tube: South Kensington.
Map.
The TatesThe
Tate Modern now gets all attention but the original,
Tate Britain, is still great. Free admission.
Tube:Pimlico.
Map .
Tube: Mansion House [and cross the bridge on the wobbly bridge].
Map.
The Courtauld GalleryUnlike other museums you do have to pay to get in, but the
Courtauld has three great things going for it: it's not very big [so there's no traipsing around], the quality of work is great, it's doesn't get very busy.
Tube: Covent Garden, Temple, Charing Cross.
Map.
The Saatchi GalleryCharles Saatchi gets plenty of bad press for his role in the art world but he can't be all bad as he's set up a great
gallery showing cutting edge work. As it's free it can get busy.
Tube: Sloane Square.
Map
Whitechapel GalleryYou have to pay to get into the main exhibition
here, but it's a very popular exhibition space for good reason. Their latest show, of Thomas Struth photos, is excellent, and runs until Sept 16th.
Tube: Aldegate East.
Map.

Other places that might interest you:
The Serpentine GalleryThe British MuseumFold GalleryDanielle ArnaudRoom
One for the kids:
The Natural History Museum:I used to love going to this
museum as a kid because it had row upon row of insects. That said, it wasn't particularly child-friendly as many modern museums now are. It is now, though. Traditionalists like myself might complain about silly moving dinasaurs, but the changes which are educational and often interactive, are obviously a huge with kids.
Natural History Museum