Recommended places in London/England
Phil Taylor

London
If I had to choose only one place in London to visit, I'd say "Sir John Soane's Museum", a little know gem. You can find it near Holborn tube station, at number 13 Lincoln's Inn Fields. It's free, and open Tuesday-Saturday.
The building used to be the house of Sir John Soane, one of England's greatest architects. Throughout his life, Soane collected an amazing range of things from all around the world, opened his house as a museum in 1833 and said that it should always remain open as it was even after his death. It might still look like a private house, and in fact you'll have to ring the doorbell to get in, but inside you'll find a unique museum packed with paintings, sculptures and artifacts from England, Italy, Greece, Asia....there's even an ancient Egyptian sarcophagus in the basement.
On the first Tuesday of every month the house is open at night too - go and see it by candlelight!
http://www.soane.org/

Other places around Britain
If you are traveling outside of London, there are many small towns and cities which are not so famous, but are sometimes more interesting than the tourist spots.
Stamford is a market town 100 miles north of London, a good place to stop on your way up North. It's a popular place for filming period dramas (for example the BBC's Middlemarch) as it very well preserved, with most buildings in the town centre built of stone. It is a good place to see cobbled streets and old stone churches without many tourists around. You can try Real Ale and afternoon teas at the many pubs and tea-rooms and buy authentic antiques and souvenirs.
http://www.stamford.co.uk

Wells is a small city in Somerset, which you should visit if you are in south-west England. It has a very beautiful cathedral, a ruined palace and "Vicar's Close" which is the oldest planned street in England. There are also markets in the old Market Place twice a week which are great for souvenir shopping. The city is called Wells because there are natural springs here (unfortunately not hot) and in fact you can see spring water running down the sides of the main street!
http://www.wells-uk.com/