Most museums and many other places of interest are free. Not free: Westminster Abbey, St. Paul's Cathedral, Tower of London, Buckingham Palace (if it's open). I think the Tower is borderline as a value proposition, but the Abbey and St. Paul's are well worth it. Skip Buck House unless you have a major royal fetish. Tourist attractions that are really flashy are probably really crap - go for the stuff that looks authentic and not brash or shouty.
This is why I think the bus tour is a good first day thing; because you can hop off and on, and see stuff as you go, but also make some notes to revisit things that catch your eye and you want to devote more time.
Taking a river boat along the Thames (pronounced "Tems") is fun too if you get a boat with commentary. You can pay with your Oyster card, and you get to see some interesting things with (hopefully) an entertaining guide. They run from Westminster Bridge (
"Look kids! Big Ben...parliament.") to the Tower, so you can use it to commute from one to the other.
Do grab a pint and a sandwich in
every any pub you come across. Just look for anything that looks cozy and authentic (and not shouty). The Lamb in Leadenhall Market, if you find yourself in the City of London, is interesting not only because the Market was a set for Harry Potter (it's Diagon Ally) but you will have some great people watching - namely the peacocks of Lloyd's of London (also Young's isn't a bad pint). Speaking of movie references, the Lloyd's building was used in Guardians of the Galaxy, as a part of Xandar's capital city that was being dive-bombed by the Accuser's raiders.
Also in the City: the Gerkin, the Cheesegrater, the Walkie Talkie and the Shard; the latter having a viewing deck at its very peak.