Hello, London parent here.
Definitely get the regular bus rather than spending out on the open top one. You can just use your bank card to pay your fare on every bus (and the tube).
If you're going to the Natural History or Science Museums, head there for opening time to do the popular bits (dinosaurs etc) before the crowds build up. You could tie this in with Harrods/Kings Road too.
For a view, the Skygarden is free, tickets are released for booking every week, for up to 3 weeks in advance. Spitalfields/Brick Lane area are interesting to walk around.
The view from the top of the Tate Modern, also free entry, are also good. My DD (7) enjoys the Tate itself too.
The Southbank is well worth a visit even though it is very touristy and gets busy. We come into Blackfriars and the walk from there to Waterloo is fun, with street performers etc. The Wahaca by the South Bank Centre is a favourite lunch spot of our kids.
The walk the other way to London Bridge is good too, past the Globe Theatre etc., and Borough Market is a great lunch destination.
Greenwich is lovely for a day out, get the Thames Clipper and visit the Cutty Sark, Maritime Museum etc. Greenwich Market for lunch on the go or there are also some gorgeous pubs such as the Trafalgar Tavern.
We really like London Zoo, and Regents Park/Primrose Hill are lovely for a walk.
Chinatown is a fun experience for lunch if your kids are more adventurous eaters. Very close to Oxford St if you're doing that.
Enjoy! It's a very child friendly city to mooch around.