Ooh, I'm so excited for you - London is one of my favorite cities! There's so much to do and see, and the Tube will take you anywhere you need to go. I highly, highly, HIGHLY recommend taking along a
Rick Steves tour book. I spent a summer in England, took another trip back to London, plus two trips to Paris with Rick at my side, and he has never steered me wrong.
Ok, enough plugging for that. Here are some things I love to do in London (some have already been mentioned):
Westminister Abbey - SO MUCH to see in here. Don't fly through - take the time to talk to the curators...they're very knowledgable and can tell you all sorts of neat stuff you never knew existed. I would especially ask them questions about the room where the Knights of Bath meet (sorry - can't remember the name of the room). They'll tell you all about the carvings on the pews (including some of the naughty ones) and what the different flags mean. I could easily spend two hours at Westminster.
Cabinet War Rooms - If you're into history, this is a really neat place. It was from these rooms that Churchill led his country during WWII.
Imperial War Museum - not as bellicose as it sounds. I loved it, but I was a history major, so this sort of thing is right up my alley. I didn't get a chance to see the Blitz exhibit, but I heard it's very good.
Harrods - Check out the food halls on the bottom level. The quantity and quality of food available will blow your mind. To be honest, I think the rest of the store is over-rated, although the Egyptian atrium is pretty cool from an architectual standpoint.
Fortum & Masons - Beautiful old department store. Can't really comment on it too much, as I just ran in here to buy some tea. The area around it is very posh - great for window shopping.
Harvey Nicholas - Now this store, I like. It's not as touristy as Harrod's and they sell some really unique things. If you really want to buy somethng at a big department store, stop by Harvey Nix.
St Paul's Cathedral - This is where Princess Diana and Prince Charles got married. If you have the time, climb up and walk around the dome. Also, visit the American Cathedral at the back of the church. Find your state flag in the stained glass window.
Piccadilly Circus - What Times Square is to New York, Piccadilly is to London. Stand in the middle and have someone take your picture.
Pubs - You can NOT go to London without going to a pub! Find one that isn't in the middle of tourist central, and order something besides the fish and chips. Above all, do not order American beer! You didn't fly all that way to have a Budweiser! Ask the bartender what his favorite is, and give it a try. If you don't like beer, have a cider.
Covent Gardens - Great for people watching, and even better for theatre. Go to the official ticket theatre ticket booth in Leister (sp?) Square (beware of copy-cats) and score some cheap seats for a show that night.
Speaker's Corner in Hyde Park - Go here on a Sunday afternoon and listen to democracy in action. It's like walking through a market of ideas - people of all ages get up on little ladders (or boxes, or benches) and start telling anyone who will listen what they think. Very, very interesting.
Tower of London - ogle the jewels, see the beefeaters.
Kew Gardens - if you have a green thumb, definitely visit Kew. I'm not really interested in gardening, but I found it enjoyable.
If you're looking for a good restaurant, I like Maggie Jones in Notting Hill. They serve traditional English food, but the staff is really young and slightly crazy. We went there on New Year's Eve and had a blast.
Also, make it a point to have afternoon tea at least once. Don't ask for high tea - that's later in the day, and doesn't come with the yummy meal that afternoon tea has.
If you have some time for day trips, I recommend Oxford and Bath. Both cities have a lot of history and a lot of beautiful buildings. If you go to Oxford, don't ask a local where the university is - they'll look at you like you're crazy. The main part of town is made up of university building mixed with commerical and residential buildings. It's nothing like our universities in the States.
I'm going to stop here...I think that's enough for one posting.
Of course, there are many other things to see, like the British Museum, Museum of London, Victoria and Albert Museum ("the V and A"), the London Eye, Trafalger Square, St Martin-in-the-fields, and on and on. Have a wonderful time, and give us a full report when you get back!