Get a good street map, such as an A-Z, and study it In advance, to see what places are near to each other and work out routes. Also study the Tube map and see where Tube stations are on the street map.
Tube is best for covering longer distances quickly. Bus or walking is best for shorter distances and seeing London.
In the City, St Paul's is the big attraction and really needs most of a day to do it justice.
There are many other Wren churches in the City, all worth a visit, and free to enter, though donations are welcome.
Also free are the Guildhall Art Gallery and Roman amphitheatre.
If you like walking, have time, and are able, the walk east from the Tower through Wapping and along the river is interesting. Stop off for lunch at the Prospect of Whitby.
The Museum of London Docklands is also worth a visit. You can go there on the DLR.
Be warned that the South Ken museums and the British Museum are likely to be very busy at any time. It's best to plan to see a few specific things, rather than try to do the whole museum.
If you're in the Euston area, the British Library is worth a visit. They usually have a couple of free exhibitions.
Be aware that pickpockets and bag dippers operate in busy tourist areas. Take care to keep your bag zipped up and don't have anything very valuable on view. In a cafe, don't put your bag on the floor or hang it on the back of your chair.
You can spend a lifetime seeing the sights of London and still not see all of it. hope you'll be inspired to revisit often!