Yep, Number 11 bus is great. You can get that from anywhere along the Kings Road and it will take you up past Parliament Square, Trafalgar Sq, The Strand, St. Pauls etc.
Science Museum
Covent Garden - lots of street entertainers to see, and also on the South Bank between Westminster and Waterloo Bridges.
Diana Memorial Park in Kensington Gardens, which is just up Queens Gate directly from the Natural History Museum.
There is usually something or other going on around Trafalgar Square, and from there you can wander up to Leicester Square and into Chinatown and Covent Garden.
River boat from Westminster Pier to Greenwich will show you lots too. I'm not sure of the cost, but I don't think it's lots and lots as it's just another commuter way of getting about town, rather than a vastly overpriced tourist trap.
Battersea Park has lots to do and you can get the number 49 pretty much past it from South Ken (although you will have to get off just over Battersea Bridge and walk down Parkgate Road and into the Albert Bridge Road entrance of the park (6 or 7 minute walk).