The world is your oyster in this town mate but be prepared to pay through the nose if you want quality.
A few tips to keep the cost down:
Some of the best museums and art galleries in the world. Free admission. If you are in the tourist areas, think about the National Gallery or the National Portrait Gallery on Trafalgar Square. The British Museum is on Russell Square, the Natural History Museum and the Science Museum are down in South Kensington. All are brilliant if you like that sort of thing.
More quality free entertainment is available hanging around Covent Garden. There is a lovely bustle to the place, some twee shops (hide your wallet) and there are plenty of street entertainers who are usually good for a giggle. be prepared to put a couple of quid in a box at the end of their show.
The tourist sights can be done quite cheaply if that's your thing. I would get an oyster card and plan your route around
On the buses - get on the top deck, and a front row view of some of the most iconic landmarks in the world is yours for a quid.
A stroll across one of the main river bridges - the millennium Walkway, Waterloo bridge, Westminster bridge - is quite memorable, especially at sunset. Somebody has mentioned the parks which are fine in nice weather and worth a look but I never find them to be massively interesting in terms of things going on in them. If you are down during the week the City is moderately interesting - mansion house, St Paul's (it costs to get in) BAnk of England, monument. Visiting the law courts in the old bailey or the Strand is interesting for some but watch out in case all those briefs and judges and coppers make you nervous mate.
If the weather is reasonable, just wandering about is nice. Plan your route though. To give you an idea what I mean, If you start at Oxford Circus and go south to regent street, a gentle meander past some very plush shops (hide wallet again) will take you down to Piccadilly circus. From there take Coventry street along to Leicester square. Wander along Shaftesbury avenue through theatre land. Then cut across to Covent garden. You can make that journey a 20 minute dash or an hour and a half stroll, taking some interesting sights in along the way.
don't drink beer anywhere south of Watford unless you really don't like having money. That said, water is not much cheaper in some places.
Finally, as it's you kev I thought i'd pass on this little message from Mrs London:
"any twat off bluemoon turns up on my doorstep, I'll set the dog on them."