The best ways to travel around the UK by mass transport are train and bus.
Train
The train network is very extensive. Your best source of information, including fares and timetable, for trains will either be the National Rail site, Trainline or the Cambridge train station. The train station is located on Station Road, off of Regent St. From City Centre, it is a 15-20 minute walk or a 5 minute bus ride. Tickets are to be purchased prior to travel at the station's ticket window or self serve kiosks. As a student, you are eligible for a Young Person's Railcard, which gives you 1/3 off most fares within the UK. You can purchase a Railcard online or at the Cambridge train station with a form of ID and a passport photo (they make it for you on the spot). The investment pays for itself in a few trips. Typical options are same-day return (cheapest), and open return (most expensive).
National Rail http://www.nationalrail.co.uk/
Young Person's Railcard http://www.16-25railcard.co.uk/
Trainline provides heavily discounted tickets. The only stipulation is that you have to go at the time written on your ticket whereas National Rail tickets are usually valid for anytime during the day. http://trainline.co.uk
Bus
If there is a town that you can't get to by train, you will probably be able to get there by coach. You can also take a coach right to the center of London (Victoria Station). National Express is the major provider of coach services; you can find time tables and fares on their website or at their office near the bus station on Drummer Street. Consider buying a Student Coachcard (called an NX2 card), which will save you up to 30% on ticket purchases. If you're up for the longer bus rides (instead of trains), MegaBus can save you bundles on traveling within Great Britain (www.megabus.com).
National Express http://www.nationalexpress.com/neh.cfm
Student Coachcard http://www.nationalexpress.com/waystosave/young-persons-coachcard.aspx
Cars
If you plan to do British sightseeing that takes you outside of the major cities (e.g. Stonehenge, Stratford upon Avon, Warwick Castle, or the British countryside), you may find that car is the most logical travel option. A map would be a wise choice, but you'll find that road signs are informative and plentiful (even away from cities).
Eurolines might be able to get you to places in Europe and back for cheaper than flights if you're up for the long rides (http://www.eurolines.com/).