I lived 2 hours outside London for 6 years. Here's my recommendations that should help you google and plan a bit more.
I didn't see how long you're staying so I only focused on things inside the London Bubble. DO NOT travel by taxi/uber/car in the city. Public transportation is much cheaper and often much faster. Mayfair is a great centrally located area.
I don't recommend a visitor oyster card anymore. You can pay via tap to pay if your card/bank/phone supports it on all public transportation.
Must do:
Natural History Museum
Victoria & Albert Museum
British Museum
St. Paul's Cathedral
Borough Market
Tower of London & Tower Bridge
Westminster Cathedral
Churchill War Rooms
Big Ben/Houses of Parliament
Buckingham Palace (only open for interior tours during specific portions of the year)
Trafalgar Square & National Gallery
Kensington Palace
Nice, but not necessary:
Imperial War Museum
Tate Museum of Modern Art
See a play (plenty of cheap tickets day of if you don't care what you go to see)
Have tea time (all of them have pros/cons. Go with whatever fits your budget/time requirements)
Wander through one of the many large parks
Waste of time:
Benjamin Franklin House
London Eye
Hop on Hop Off Buses (City buses go everywhere, just use Google Maps with the transit option). Although I may recommend a Hop on/Off Bus if you want a day of just sight seeing the city and don't have any desire to go into many of the historic sites I listed in the Must do.
If you would like to leave London, I can give you a few great recommendations that are relatively easy to get to and/or can be seen as part of a day long whirlwind bus tour.