If you intend to travel outside London to other UK towns and Cities then the rail network is both modern and quick.
Look to book your tickets online. The more notice you can give the cheaper your tickets will be. In addition if you are looking to spend any length of time here look to invest in a Railcard these will save you money if you plan to do a lot of travelling.
Must see cities in the UK for me would be:
Bath, Stratford, Edinburgh. There are of course many other options for visiting ideas you can get from in here if you let people know what your interests are!!