Three days in London
Try to get in to London in the morning. If traveling across many time-zones, try to avoid the urge to sleep your jet-lag away — this is best done in a cheaper city! At the earliest opportunity, get an Oyster card from the vending machines or from a newsstand. This will save you about half the cost of a one-way ticket on the tube and buses.
Drop your bags at the hotel, and you can start immediately. Bus tours are always a good way to get a feel of the city. The Original Tour ticket costs £23 for adults; it is a hop-on/off system, good for forty-eight hours. Three walking tours and a river cruise are thrown in.
Explore Trafalgar Square, taking the obligatory photographs. When ready, you can start the bus tour. While the bus tour allows for you to alight at any stop, I find it preferable to get an overview of the city first. The tour is short, lasting only 90 minutes. You can come back to any stop of interest either by another similar bus, or by public transportation. Unfortunately, many of the tour guides, while informative, are not quite as entertaining as those you find in New York City.
Try to arrive at the Tower of London at about 2-3 pm. Wait for one of the walking tours conducted by the Yeoman Warders, and follow the crowd as you get yelled at by a former British Army sergeant-major. After the end of his commentary at the chapel, you can explore the rest of the grounds.
A quick bite at the restaurants near the Tower of London, and at about 7 pm, the Jack the Ripper walking tour begins at Tower Hill underground station, ending at about 9.30 pm. This concludes the first day.
The Change of Guard at Buckingham Palace is every other day. If you can, go and take a look at 11 am. The entire ceremony lasts about three hours, much of it behind the fence. A quick look of the guards marching in and out, followed by the horse guards is sufficient.
After this, take a stroll down the park toward Westminster. Food can be purchased from a supermarket near the pier. Once done, get to the pier and on a riverboat cruise, seeing the sights of the Thames River. Get off at Greenwich (1 hour). Greenwich is a quiet small town, and the pace of life is noticeably slower. The Royal Observatory is a bit of a walk (30 min each way), but when you get there, you can see the Greenwich Meridian, and set your wristwatch by the displayed clocks. The museum is worth a visit as well. Don't bother taking the cruise back; you can use the Docklands Light Rail to get back to Tower Hill.
If you are inclined, you can have a comparatively cheap dinner at Chinatown (Leicester Square underground), followed by a show at the West End theaters. Be sure to purchase your ticket either online, or from the numerous half-price ticket booths around Leicester and Trafalgar squares earlier in the day. Make sure you receive your actual admission ticket.
Check out of the hotel, and leave your luggage. Depending on your interests, you can go shopping on Oxford Street, or you can visit the many museums and galleries. Get to the airport/railway station early, and wave goodbye to expensive London.
A suggested next stop is Paris in three hours, by Eurostar. Discounted tickets tend to be in the evenings. They start at US$60 per head; book early online.