Hey guys, sorry I'm late to the party.
I've been to London twice in recent years (meaning since 1999) and have some suggestions:
1. Do go to the Tower, but do get there at opening, and get on the first or second tour with the beefeaters. There will be a somewhat more manageable mob, and will leave you a good deal of time afterwards to poke around on your own. When I was there, the beefeaters also offered lectures in the afternoons. I went to one on the history of imprisonment and torture in England, which was fascinating and horrifying.
2. I found pretty much througout England that the guided tours offered at sites are highly worthwhile. For a couple pounds or something like that, it makes the visit to the place a lot more rewarding. I would say that the tours at the Tower and at Westminster Abbey are musts (the Abbey one takes you to places that are not open to the non-touring public). I didn't try the ones at the museums, so not sure about thoise, but I highly recommend them for historical sites.
3. Many (most?) of the museums have no admission charge, which makes it easy to see them in more than one visit. I saw the British Museum over parts of three different days when I was there. Whenever you find yourself with a spare two hours or so, you can pop into a museum.
4. As GD mentioned, the half-price ticket booth is an excellent option, though when I was there it was in Leicester square, not Trafalgar. Perhaps it moved? I went to a different show essentially every night I was in England, and they had tickets for just baout everything available there. I got tickets there the Royal Shakespeare doing Measure for Measure which was the best production I have seen of anything, anywhere.
5. The one sort of out of the way place I got to and recommend is the Old Operating Theatre Museum (
http://www.thegarret.org.uk/oot.htm). It is an actual operating theater, dating from 1822, that has been turned into a museum about pre-Victorian medicine. Also fascinating and horrifying (actually, those are recurrent themes in England).
6. Day trips could be a lot of fun as well. Canterbury and Oxford would be manageable day trips from London, and well worth while. Cambridge and Stratford might be as well (I haven't been to either). Stonehenge, I think is a little farther away, and I'm not sure how day-trippable it is. You can check my geography on this. When I went there, I had a rental car and I got to Stonehenge in the morning as it first opened. I was the only one there for a while (two or three people showed up before I left), and there was a lot of low-hanging fog around and in those conditions being with the stones was a fairly awesome experience. Then I was able to drive to Avebury and see the stones there as well. Anyway, what I'm saying is that I'm not sure a bus from London would get you to Stonehenge in the right time or frame of mind to appreciate it fully. Maybe I'd leave that for the next visit. Bath is also surpassingly lovely, but I would want more than part of a day there. I also recommend York and Edinburgh, as long as we're planning your next trip.
7. It doesn't get as uch attention as some others, but I really enjoyed the Museum of London (which covers the history of London).
Anyway, that's my several cents worth.