When i lived in London I had a little loop i used to take visitors on. It goes past some of the lesser regarded landmarks but i think it's a convenient route.

You start at st pauls cathedral on the north side of the river and cross over millennium bridge to the south. On the south bank is the tate modern, it's a huge modern art gallery in an old power station. Free to enter if you like art. Follow the river to the east and you'll go past the globe theatre which is a recreation of the theatre Shakespeare wrote his plays for and the only thatched building allowed in London. Keep going east and you'll pass the location of the clink which was a medieval prison and the ruins of Winchester palace. Next you come to the Golden hinde which is a recreation of the first English ship to circumnavigate the earth. After that is southwark cathedral and borough market which is a great place for lunch if a bit pricey. You're very near the shard here which is the tallest building in the UK. You'll go past London bridge which is famous not interesting. If you look over the river to the north you should be able to see the monument to the great fire of London. Keep following the river East and you go past the hms belfast which is a ww2 warship. After that is a plaza outside what used to be London city hall. Next is tower bridge which is the famous one on all the postcards. Cross over the bridge back to the north bank and you're right next to the tower of London which is a castle where the crown jewels are held and where monarchs used to keep political prisoners.

From here there used to be a bus route that still ran the old 60s routemaster double deckers but sadly it stopped. You can get a much more modern bus from here, the 15 westbound, it goes back to st pauls where you started but if you stay on it it goes down some old historic streets like fleet street (famous for being the home of the first printing press in England and its association with newspapers and printing) and the strand. It terminates at charing cross which is near covent garden and trafalgar square where there is loads of historic stuff, museums, theatres and attractions.

Other good stuff: If you're into military history the churchill war rooms are incredible

The huntarian museum at the royal college of surgeons is a collection of old medical samples, curiosities and historical teaching aids

The grant museum of zoology has some really weird and interesting natural history specimens and its quite small so you can go round in probably less than an hour

The museum of London is underrated and is good for getting an idea of the different eras of London

That's what I'd recommend.