I lived in London all my life and yes, London can be terribly expensive especially in Central but it is possible to keep the expenditure down if you know where you are going and avoid shopping in the High Street stores such as Oxford Street, Regent Street or Harrods, although window shopping is fine

. I'll just point out a few places worth visiting.
Camden is a place definately worth checking out. It has several large markets of a style very unique to Camden, not to mention about the music and bars at that place.
Soho is a small area between London Chinatown and Oxford Street. It has quite a few bars, karaoke places and resturant. It is also apparantly the LGBT area of London. A few premises actually fly the rainbow flag over the door.
Covent Garden, another market place with quite a bit of history. There are usually street performers to keep you entertained as well whilst you sit down and have something to eat. The London Transport Museum is also nearby.
If you have some time in the day, check out the Docklands area, where the finance district is. There are a few modern shopping malls and alot of water surrounding the old docks. If you go across the river, you'll reach the Royal Observatory, Greenwich Palace and a lot of green space. Excellent for a day stroll to get out of the busy city life and enjoy the nature.
If you want more green space, check out Hampstead Heath, an enourmous green space where you can view the London skyline if you stand on the highest point of the Heath- Parliament Hill. The view is awesome in the evenings.
Of course, there is the usual tourist attractions such as the Big Ben, Houses of Parliament, London Eye, Trafalgar Square, Buckingham Palace, Hyde park. I won't go into details about these as they are all over the internet.
And lastly... If you are planning to travel alot, do yourself a favour and get a Pay as you go Oyster Card. This is London's universal smartcard ticket which allows you to travel on the Underground, buses, trains, Docklands Light Railway and the tram. If you top up around £20 credit, it should get you around for 3 days. Don't pay the cash fare as it's a rip off. I read on the Evening Standards paper on tuesday and there are souvenir editions of the Oyster Cards with a picture of Prince Willian and Kate! May be worth getting one of those whilst you are at it.
When you have the Oyster card, just make sure to tap your card on the Oyster readers by the gate when you enter and leave the station and the fare will be deducted automatically. If there are no gates (i.e. the Docklands Light Railway), there will be a Oyster reader by the entrance, tap you card and if its OK, you get 1 beep and a green light. If it not, you will get 2 lower beeps and a red light... the usual reasons are either not enough money on the card, or that the reader has failed to read your card. You will need to read the LCD on the reader to find out the reason... yes, they DO check tickets on the trains so I probably wouldn't want to risk jumping on with an unvalidated card! On the bus, just tap it on the reader as you board.
Hope this helps!
Kairi (A life-long Londoner)