I went last year with friends, and 7-10 days is NOT enough, just for London! LOL
Bring GOOD walking shoes, you'll be doing A LOT of walking! Even with taking the Tube (subway) you spend a lot of time on your feet and on stairs! There's tons of stairs, but no apparent hills! LOL
We did the hop on hop off bus tour and it was well worth the money (£20 or so for the single day one. (You can get a two day pass as well) Takes about 2 hours if you don't hop off anywhere, but is well worth it to hit all the "highlights" with a tour guide on day one and decide where you want to go the next day.) It also included a bit of a water hop on hop off tour. You can see all the sights from the Thames, but they aren't all narrated. (Ours was not, but we had fun anyway!) Start your tour at Trafalgar Square, you can buy tickets at some of the tourists stands and they seem to have people roaming the streets looking for tourists to buy tickets. There are 3 lines that sell them. We went with the "Original" tour, (I think it was the Red Line) but the other ones offered tours in other languages. They also provided headsets for translation if you wanted that.
Don't know anything about hotels, we rented a flat in the west end of London, and traveled by subway and bus (and more walking!) for the whole trip. Our only taxi's were from the airport and back again.
If you want to see a show (Phantom of the Opera, Wicked, and many many more) you can book ahead online or you can go to a ticket outlet in Leicaster (sp) square and buy them for a discounted price. We missed out on Phantom however because we didn't book in advance and happened to be in town when some huge U.S.A/UK sports thing was going on.
We saw the Tower of London (take the inlcuded 30 minute tour offered there, with the Beefeaters, on the grounds, before you go inside. It's fun and they are quite amusing and quite knowledgeable. And LOUD...They do the whole 30 minute tour, outdoors with a crowd of about 100 and they yell the whole time, but they are very clear! I'm glad we went on that. Then go indoors and take the guided headset tour. (They are free) Sooooo much to see and so much history. You'll see exactly where Henry the VIII beheaded Anne Boleyn, and Katherine Howard, among many, as well as lots of other very interesting bits of information that goes right up til WW2. (They were preparing the Tower to be used by Adolf Hitler should they capture him.)
We also took the train out to Buckingham Palace and had another great time there. If you see any docents (not sure what they are called there) ask them questions. They seem quite excited to have people who are interested to talk to and can tell you all kinds of interesting things that are not on the "headset tour". Like the fact that Henry VIII's grave is not "exactly" where it is marked. We were looking at it, and the docent was telling us about it, but it turns out that after they x-rayed the area they found the grave was a few feet away and we were actually STANDING ON TOP OF IT, as we spoke! (And don't ask about the movie "The King's Speech" the docent seemed quite insulted that someone would make a "film" about Queen Elizabeth's father, King George...LOL)
I never made it to the Victoria and Albert museum, but will be doing that this time (I'm going back in August).
The British Museum was also fabulous! I saw the Rosetta Stone in person, along with an amazing ancient Egyptian display and sooooo much more! Another whole day could have been spent there. (Actually a few days!)
Again, so many things you could do, I can't even begin to tell you. What kind of things are you looking to do? Shopping, eating, sightseeing?