Some bus companies travel very cheaply to Europe, for example Megabus.
Train travel tends to be more expensive although you can obtain discount as a student on a number of travel tickets.
Bus? Train? Plane? This Australian website has a useful summary and advice for travel on a budget
If you use a budget airline check where the airport is, as it can sometimes be quite far from where you'd like to go. You can compare costs of flights online with price comparison sites such as Skyscanner.
If you live in London, you may want to join the International Student House Travel Club which organises discounted trips inside and outside the UK. Your institution does not need to be a member for you to join.
Your Students' Union may also run discounted or cheap trips or be able to advise you how to save money on travel. And ask other students.
Before you travel in Europe, you may want to buy an International Student Identity Card which can give you benefits and discounts across Europe and beyond.
The Budget Traveller website has 75 tips for budget travel in Europe