In terms of fitting things into a single day, you'll probably want to check the opening hours of places you're interested in, as it's a Sunday. Honestly somewhere like the British Museum or the National Gallery you could probably spend the whole day, but maybe picking one thing for the morning and one for the afternoon would enable you to balance things out a bit. You might also want to plan to visit a park or something else that's always open earlier in the morning or later in the afternoon when other stuff starts to close, or make a plan for good weather and another one for bad weather (always a good idea in the UK!). Check out where things you're interested in are in relation to one another (both in the physical world and in terms of the Underground, which is much easier to navigate than the buses for a London newbie) in case you can take in one thing on the way to another. And get a map!

You can't walk uninterrupted along the Thames- it is still a working river, after all- but there are large stretches where you can walk. One of the nicest is South Bank- you can visit the Tate Modern then stroll along the river front where there are some nice (if overpriced) shops and restaurants, a big second hand book stall, lots of street performers, and usually an ice cream van or two.