Yes lots of galleries & museums are free (except for temp exhibitions):
Museum of London
Victoria and Albert Museum
National Portrait Gallery (also has free film screenings and free music events)
Theatre Museum
Photographers Gallery
Serpentine Gallery
etc etc - there are lots of free smaller ones
Almost free: ICA (Institute for Contemporary Arts) - you pay a day membership of £2 I think.
A great way to see some of the sights on a clear sunny day is a boat trip along the Thames - most will cost you about £6 and leave from Embankment or Westminster piers in town and go along the river to places like Greenwich or Richmond. There are also canal boat rides from Camden to Little Venice, similar price.
Some places you would expect to be free are NOT - Westminster Abbey is £5 I think and St Pauls Cathedral is £7.50 so be warned!
Highgate Cemetary is worth a visit, it is £2 to get in (if you don't have family buried there) and you can see graves of lots of notables including most famously Karl Marx.
Also free of course are the parks and open spaces: Hyde Park, Green Park, Kensington Gdns, Regents Park, Primrose Hill, Hampstead Heath, Richmond Park, Battersea Park.