It might sound like a tourist trap, but I highly recommend the open top sightseeing bus. I would do that on the first day.
You get to see a lot of the typical and most famous sights in London.
I find a lot of them are nice to "have seen them", but you don't actually need or want to spend much time there.
If anything interests you specifically, you can either get off the bus (the hop on and off ones) or you simply return on another day for a more in depth visit.
You get a feel for the "lay of the land" (even though you will probably be stuck in traffic a bit) - much more so than when you travel underground on the tube (which of course is great to get fast from a to b)
A lot of museums are free (only specific exhibitions cost)
For a cheap meal, (fixed price menu, 2 courses £17) in a spectacular room, go to Brasserie Zedel at Piccadilly Circus. It looks like a little French bistro on ground level. Ignore that, walk through it and go downstairs and you find yourself in a vast marble palace. You might need to book.
Other good food place is Mercato Mayfair
Or visit Brixton Market (not the street market, but the two covered market halls), 3 minutes from Brixton tube, for the choice of tons of little restaurants
Yes, to Citymapper app!
Pre-book Skygarden. It's free and wonderful
I assume you arrive at Heathrow.
Get an Oystercard (you but them at the Heathrow tube stop) and add money (start with something like £30) and you can use it on the tube, bus, trains and boats. There's a daily cap which means you won't get charged more than the cost of day travel card.
Be aware that there are three options to get from Heathrow into the centre
I looked up the costs from Heathrow to Dalston
- Heathrow express (£28.50 takes 60 minutes)
- Elizabeth Line tube line (£13.90, 73 minutes)
- Piccadilly line (£5.80, 84 minutes)
Have a wonderful time