I took the Eurostar from London to Paris recently. High recommend it, very fast and only takes just over 2 hours. Price can be the same as a plane or cheaper, just depends when you book.
Plenty to see and do in London. Once you get a tube map you will enjoy navigating your way round! As already mentioned make sure you get an oyster card the fares are cheaper.
Boose is cheap, esspecially from all the little corner stores. Oh and if you like mexican check out Chimi Changas