Things I enjoy doing : 

The markets. A particular favourite is Borough Market (upmarket food) on a Saturday lunchtime. Turn up, grab a coffee from Monmouth's, wander round, have some lunch from one of the stalls (chorizo from Brindisa, raclette from Neal's Yard, burger from the Ginger Pig or a pie from Pieminister, for example).

The Thames. The South Bank between Tower and Westminster Bridges is a great place just to wander. There's quite often events on around Butler's Wharf on a weekend, too. There's loads of attractions on the way, too, and the espresso bar in the Tate modern is great fun.
 

River Boats. There's tour boats, but that's not something I tend to enjoy. But there's also a number of commuter services sprung up along the river detailed here. The Greenwich-Savoy section is particularly fun, and the Richmond-Hampton Court is easily the prettiest way to reach the palace!