As an American who has visited the UK about six times (three of those times within the past two years), I suggest the following places for visits (my top 10--12 if you count the two nightlife suggestions at the end):
(1) London--of course (Parliament, Hyde Park, Buckingham Palace, Trafalgar Square, theatres up by Leicester Square, Covent Garden, Tower of London, Downing Street, and the shopping districts of Mayfair, Oxford Street, Picadilly Circus, and High Street Kensington). Also, the museums here are superb--the British Museum in particular being the crown jewel. They also have art museums, a natural history museum, a World War II museum, and even a Sherlock Holmes museum
(2) Bath--a gorgeous old Roman town with great open squares and a good shopping district, and it is best known for its Roman Baths (hence the name). It also has an old Jane Austin shop/museum. Just southeast of town about an hour is the Longleat house. In the gardens is the largest hedge maze in the world--definitely worth a visit.
(3) Warwick Castle--best preserved castle in the UK
(4) Stratford-upon-Avon--Shakespeare's old home town. The central area is all touristy, so best avoid it, but try to get to Anne Hathaway's cottage and Shakespeare's old gardens slightly outside of the main downtown area.
(5) York--the best preserved medieval town in England, and also the most haunted, so they say
(6) Edinburgh--the castle and the city are interesting. Also, the area immediately surrounding is great--especially if you like to golf or visit distilleries
(7) the castles of Wales--lots of old castles and abbeys out here in a really beautiful area of the countryside. The largest castle, Caerphilly, in the south is especially impressive.
(8) Stonehedge--just to say you were there and see the unusual site
(9) Cambridge--very unique college town--much preferable to Oxford, in my view--although that one is also okay; and
(10) The Cotswolds--this area lies about one to two hours west of Oxford and if you take the small roads, you'll find some countryside that really makes you think about nature and also make you feel like you are back living in the better part of the 1600s
Lastly, I should mention Manchester and some small towns around it--everyone who lives in Manchester loves being there, and both in that town and in the town of Milton Keyes, the nightlife is really "happening"--same as in London--so good places to go to if you are single--and okay for a young married couple, too.