Skip Madame Tussaud’s–way overpriced and the only real novelty from American wax museums is they have a lot of royalty you don’t know and are often free-standing in non-dioramas so you can get up real close (Oooooo! :rolleyes: )
Walking across Tower Bridge is cool, but the Tower Bridge Experience is kinda cheesy; it’s interesting on the history of the bridge and you do get to go along the upper walkway, but there are touristy animatronics that seem really out of place in most London sites.
The Buckingham Palace changing of the guard is way overrated, especially since you have to go really early if you want to have any hope of seeing anything, which means you’ve killed a good 1.5 hours (at least) of valuable time in the middle of the day. A good alternative is the changing of the Horse Guards right across from the Banqueting House (which has Rembrandts on the ceiling). No music and less formal ceremony-business, but essentially the same thing, but with horses and no crowds.
First day, if you can, take one of the double-decker tourist buses that take you all over the city. This is a great way to get oriented to where everything is, and if you don’t make it to all the places you planned on going, you can at least say you saw it. Also, things you might not have considered may look interesting (or vice versa) so it helps narrow your choices. If you’re pressed for time, these buses are all hop-on/hop-off, so you can actually get off and see something and then resume the tour later.
Many of these bus companies also have good package deals. One I took allowed you to buy tickets for the Tower of London through them, which means you can bypass the (usually horrendous) long line. You can also get theater tickets for discounts there too if you want to catch something in the West End.
Of the things already mentioned, St. Paul’s, the Tate Modern & Old Globe, British Museum would be my first recs. The V&A, Tower of London, Hampton Court are all great but definitely take a bit more time (as do a lot of the other museums, also).
One nobody’s mentioned is The Imperial War Museum, also off of the South Bank. Very impressive and given England’s history, incredibly comprehensive.
Time Out is essential because it also lists the temporary exhibitions. There are a ton of historical and cultural walking tours (not just Jack the Ripper) and TO lists them daily as well. You should definitely go on one, especially to get a look at some of the nooks and crannies that you don’t get from the standard tourist destinations. It also is good at highlighting different market districts, so if you’re into, say, antiques, you’ll know where to go. It also provides a calendar of special events, concerts, fairs, etc.–some of them free–that you might be able to catch while you’re there. Definitely worth the few pound investment (it’s a weekly publication).
Wow, have fun!