London's public transport system is the easiest way to get around, with extensive underground ("Tube"), bus and rail networks across the city. As a student you'll be eligible for 30% off all adult travelcards and bus passes valid for a week or more if you apply for an 18+ Student Oyster card and are on a programme for 14 weeks or longer. Cycling and walking are also great ways to get to know the city; London has a popular cycle hire scheme and guided walking tours, often based on historical themes, are also available.
If you want to travel further afield, London has direct links to the rest of the UK and is a great base for further exploration - day trips to the seaside at Brighton, or Shakespeare's birthplace at Stratford-upon-Avon, for example. The Eurostar terminal at St Pancras International (a few minutes' walk from UCL) offers direct, high-speed rail connections to Paris, Brussels and Amsterdam, and London's various airports give you the opportunity to further expand your horizons.