LOVE, LOVE, LOVE London!
Best way to get around is in "the tube." They're clean and very well mapped out. You can buy daily, weekly or weekend passes. The passes require passport sized pictures, so it's a good idea to get that done and bring those with you. This is a site which can give you more info:
http://www.tfl.gov.uk/tfl/
All of the suggestions by others as to where to go are excellent. I'd add.. Westminster Abbey. It's expensive to get in, but well worth it.
The wax museum (Madam Tussaud's) is fun too. Some of the walking tours are fun-- including the ghost tour at night.
The Houses of Parliament are very interesting. You can get a guided tour from June through Sept.
Personally, I would skip "The Changing of the Guard." It goes on forever.. it's really crowded, and it's not that exciting. If you really want to go, be at Buckingham Palace no later than 11:15am.
You can find a sightseeing tour which will take you around all of the usual touristy spots. A good one is the "hop on-hop off" double decker bus. It goes all over London, and when you see one of the main sites you want to see, you just hop off, then get the next bus and "hop on" again. Here are some sites:
http://www.bigbus.co.uk/
http://www.theoriginaltour.com/
You can also find bus tours to some great spots like Stonehenge, Oxford, and Bath.
Some misc. info:
Londoners are very friendly. If you need help, by all means ask. When we were lost, we stopped and asked people.. and they would stop to draw us maps, and even walk us to our destination. At first brush, you might think they're not friendly.. but you have to approach them for help first. Once you do, you'll find they'll go out of their way to help.
Learn to speak English.

Lots of the words are different. I remember trying to buy a curling iron, and nobody knew what I was talking about. They call them curling "tongs."
Of course, you probably know a "bus" is a "coach", a "dumpster" is a "skip," an "elevator" is a "flat", the bathroom is "the loo" or the "w.c."
Of course you know what a British Pound is.. but it is often referred to in slang as "quid." So if you have 20 pounds, they might say "20 quid".
Do not say "fanny". In the USA, fanny is a casual slang word for your posterior (the English say "bum") but "fanny" it's a very vulgar word in the UK!
You don't tip the bartender. If you don't like beer, it's sometimes hard to get typical American cocktails at pubs. (Don't even try to order a Sombrero!) You can, however, get simple "rum and coke" drinks.
You usually don't tip in restaurants either. Look on your check-- it's usually already included.
If you're a chocolate lover.. go into a Woolworths and load up on Cadbury. They have versions we don't have, like my favorite "Cadbury Crunchie!"
Food can be quite pricey. Be sure to ask the locals for a good "Chip" shop and buy the fish and chips. You put vinegar and salt on both the fish and chips (french fries.) If you want to sample other traditional fare, you could eat in a pub and order a good soup or a meat pie (just don't ask what goes into them!)
You can also find a huge assortment of foreign cuisines... but again, quite pricey. If you like Dim Sum. try Royal China in Queensway. Look for Mr. Wu branches all over town for a Chinese buffet. There are lots of Indian restaurants, Sushi bars (Moshi Moshi in Liverpool Station) is excellent, but very expensive, Thia, Vietnamese, and French.
If you have an emergency.. dial 999 for police, fire or ambulance.
Have fun.. you'll have a great time!