Gosh there’s just so much to choose from… it depends what you want to see!
Regents Street / Oxford Street / Leicester Square area have the big flagship shops. Covent Garden is nice to get lunch and browse smaller interesting shops/craft market stalls.
If you want posh shops Harrods, Liberty, Fortnum & Mason are an experience.
Sightseeing - you could do a walk that takes in Buckingham Palace, Trafalgar Square, Westminster Abbey with St James’s Park in the middle of them all. Watch the changing the guard - www.changing-the-guard.com
The London Eye is a good way to see the skyline. Often see half price tickets around eg cereal boxes.
The Transport Museum in Covent Garden and the Tower of London are both good for a flavour of history.
The Tate and Shakespeare’s Globe are next door to each other if you want some culture, both impressive buildings.
Uber boats are fun to hop on and off and see some sights from the river.
Or hang around the museums - the Natural History Museum and the Science Museum are round the corner from each other so easy to group together. The V&A is fab for fashion, jewellery etc. All free!
I would work out what you want to see first then book a hotel nearby. Try to group a few things near to each other for each morning/afternoon so you don’t spend loads of time walking or on the tube.