London is an absolutely fantastic city, but it sure ain't cheap. Here are my best ideas for getting more for your money.
get a train from the airport
All the airports in London are miles and miles from the City Centre, so taxis are prohibitively expensive and car hire in London is a nightmare. They are all served by reasonable train links though, so let the train take the strain. If you are feeling particularly frugal, you could get the tube from Heathrow, although it takes forever and is not recommended if you have a lot of luggage.
get a map and walk
Many areas of the city are eminently walkable. Especially the West End and the City. For example, it takes less than five minutes to walk from Leicester Square to Covent Garden. It probably takes longer to get to the tube platform if you go on the underground. Also, if you walk around, you get a better sense of the place and the architecture.
get an oyster card and travel on public transport.
Public transport in London is pretty comprehensive, almost everywhere of interest is located close to a tube station. The cheapest way to use public transport is by getting the integrated smartcard called Oyster.
If you buy in advance, you need a £3 deposit and to pre-load it with £10. If you wait till you get there you the minimum top up is 10p, although you still need pay the £3 deposit, and you should probably start out with putting £5 on it. With an Oyster card, single zone 1 fares on the tube are £1.50 and without they are £4. The maximum you can pay in a day on the tube is also 50p cheaper than the equivalent paper ticket. Assuming you make at least two zone 1 tube journeys a day in your trip, if you want to keep the card, you break even within three days and if you don't want to, within two.
A word of warning however, the tube gets incredibly hot in the summer, and unless you are used to underground / metro / subway systems in general I wouldn't recommend that you travel in rush hour. Make sure that you stand on the right, unless you want to walk up/down the escalator at high speed, or be physically moved out of the way.
if you're out on the town, get a night bus home.
The routes are here and its much cheaper than a taxi, although it is what is commonly described as an experience.
Things to do
get almost half price west end theatre tickets.
The catch is that you need to get them on the day from the tkts booth in Leicester Square or Canary Wharf, there will be a queue, they charge a £2.50 booking fee and they can only sell the tickets available to them on the day.
In all the tube stations etc you will see free papers the most quintessential is The Metro. Its great for whats on.
All the major museums, (like the British Museum, the Natural History Museum, the Science Museum, the V&A Museum, Tate Modern, Tate Britain etc) are free for normal entry (some exhibitions may charge). These aren't just ordinary museums, for example if you want to see the best of Egypt in one place, the British museum if the place to go, you could easily spend a day in any of these places and not see everything. Do note that food at the museums whilst usually good quality, is not cheap. But its pretty easy to leave the museum, walk up the street to a sandwich shop, eat and come back.
parks and other oddities
If you're bored with London, you're bored with life. However, there are plenty of day trips from the capital, including the esteemed University at Oxford, the Royal Castle at Windsor and you can even go to Paris for the day. All of these are within your realm if you book train tickets well in advance. Try the trainline.