London is a surveillance society, with closed-circuit cameras everywhere: in the museums and theatres, on public transport, in and near office buildings, in your hotel lobby, and probably even in the public restrooms. I personally find that to be very creepy, and it has caused me to stop going to London. As others have noted, Heathrow Airport is pretty terrible, but you can take the Piccadilly Line subway into Central London quite easily. Make sure that you buy an Oyster Card at your first opportunity so that you don't pay the highest fares on the Underground.
You're going to find a laptop useful, not just for your email, but for restaurant guides, theatre tickets, and all of the usual travel info. As in the US, the expensive hotels will charge you a fortune for Internet access, and the less expensive ones will provide it for free. You just need to find and set the appropriate SSID. You will find pay WiFi at Heathrow (The Cloud is pretty good service), and that will come in handy if your flight is delayed or cancelled. You should bring an unlocked GSM mobile phone, and buy a prepaid SIM card for about $20 at Phones 4U or Carphone Warehouse. I've had good success with Virgin Mobile there. That gives you a local phone and number, and avoids the roaming and data charges of your US carrier. It's easy to find an unlocked GSM phone in the US before you travel - try Craigslist.
London theatre is outstanding, and you can often find discounted tickets at the official ticket booth in Leicester Square. Be careful about this, since there are many other ticket agencies in that neighborhood pretending to be the official one.
Others have suggested the various London tourist attractions, but I would add that you would do well to get out of London. Both Cambridge and Oxford are easily reachable in about an hour by train (Liverpool Street for Cambridge and Paddington for Oxford). It's well worth a day trip to each, and I highly recommend seeing some of the various colleges, as well as the Ashmolean Museum in Oxford.
As for shopping, prices in London (all of England, really) are so high that you're not going to want to buy a lot of stuff. In addition, one effect of globalization is that many products that you will find there are available in the US at lower prices. Of course, it's still enjoyable to wander through Harrod's, Hamley's, Liberty, and other unique London shops. Oxford Street and Regent Street stores will be nicely decorated for the holidays, which makes for great window shopping.