I had to go to London for 3 weeks due to work. During the weekend and after hours, I had plenty of time to enjoy the city and sights. The Tube (subway system) is easy to use, very cheap, and safe.
Some free things to do in the city: Go to Buckingham Palace and walk around the outside. I went to he Royal Mewes and was able to view the Coronationation Carriage, Wedding Carriage, and horses. See the changing of the guards. The guards will not show any emotion, change their facial expression, or say anything. Visit Westminster Abbey. There are lots of museums in London that are totally free. Some of my favorites are the British Museum, the Museum of London, Natural History Museum, and Victoria and Alert. Go to Harrod's department store. They have a memorial and display of the ring the Dodi Al Fayed gave Lady Diana. See Big Ben. There are lots of gardens you can view. For garden views, try Kensington Palace, Hyde Park (there is a Peter Pan statue and a Diana Memorial), St James Park, Regent's Park. See Tower Bridge. Go to Covent Gardens. There is always a lot of activities here, including street players and musicians. Go to Piccadilly Cirucs - this is kind of like New York's Times Square. Walk down South Bank, which is along the River Thames. You can see the London Eye here.
Some of the things I did that I paid for and thought were worth is was: Maddam Tussoud's. It is amazing how these wax figures are so lifelike and real. This is a one time trip only for me. I liked it, but would not do it more than once. Tower of London - a great tour. A day trip to Windsor Palace. A Jack the Ripper Tour. There are lots of these and they each give their own theory. I took the one by Ripping Yarns Walking Tours, ltd.
Food. The food here was really not so great. Make sure you do stop for Fish and Chips. I really liked Liberty Bounds pub by The Tower of London. I went to China Town and had Chinese food. That was probably the best food I had there. You can also eat in the large department stores. I tried to stick to egg and mayonnaise sandwiches (Egg Salad). I was surprised to walk in one sandwich shop and find that they charged more if you ate in, versus if you took it to go.