There have been a few mentions of transport passes/Oyster cards - these days the pricing is such that it’s most economical to pay as you go. Because of the daily cap, you end up paying less. And you can use any contactless payment method (Apple Pay or contactless credit card) to go through the gates, you don’t even need to deal with topping up an Oyster card.
Without knowing your interests, and assuming you have no mobility limitations (plan incorporates a lot of walking), for four days in London, I’d do the following (doesn’t have to be in this order except the first day but this is how I’d group the sights):
DAY 1 (I’m assuming you’re landing on this day? This is usually what I do with visitors on their first jet lagged day)
Take the tube to the Westminster stop
Walk through Westminster to see all the big landmarks (sorry that Big Ben is under scaffolding!), I think Rick Steves actually has a walking tour you can follow
If time and energy allow, Westminster Abbey - because so many historical figures are buried there, the audio guide ends up giving an overview of British History that I think is useful background
DAY 2
Take the tube to Tower Hill (if you’re energetic and up early, you could start the day with a walk up Fleet Street and through the City; depends how ambitious you are)
Tower of London (get there when it opens at 10)
Walk over Tower Bridge and then along the Thames to Borough Market for lunch
Tate Modern (modern and contemporary art; book ahead for any temporary exhibitions, which are always excellent)
Walk along the South Bank, stop at one of the places by the National Theatre for a cocktail and people-watching
Consider dinner and a show at the National Theatre or the Old Vic for evening entertainment, or alternatively take the Bakerloo line from Waterloo up to Oxford Circus for dinner and a night out in Soho
Day 3
Pick either the British Museum or the V&A depending on your interests
Spend the afternoon exploring Notting Hill and Kensington (if you pick British Museum, you can take the Central Line from Tottenham Court Road to Notting Hill Gate, explorai Notting Hill, and then walk down into Hyde Park, Kensington, and South Kensington; if you pick the V&A, you can start with South Ken and walk up through Kensington, Hyde Park, and into Notting Hill)
Do “afternoon tea” for dinner - my favourite options are the Ritz (for something classic) or Sketch (for something more whimsical), both need to be booked far in advance during peak tourist season
DAY 4:
Choose between Parliament tour (beautiful Palace of Westminster interior plus super interesting if you’re at all interested in politics and government) or National Gallery (British and European art)
Sandwiches in St James’s Park for lunch
Churchill War Rooms
Explore Covent Garden in the late afternoon, have an early dinner
West End show
For anything that you can book ahead, do. For museums, check whether there are any temporary exhibitions you want to see and book tickets. Others can comment on the best way to get last minute theatre tickets; for most shows I book in advance, or if I decide to see a West End musical last minute, I use the Today Tix app.