Even if you are in Florence for one semester, you may travel in Europe at least 5 out of the 12 or 13 weeks of your study abroad period. Some of these travel weekends will likely be in Italy, with an organized group or a school field trip where transport and accommodation are already arranged for you.
However, if you want to visit such places as London, Dublin, Paris and Barcelona, and other cities that are not part of your program or local tour providers' offerings, you are going to need to organize your own travel.
One of the best deals for students, and a fun way to explore Europe, is by staying in hostels. Compared to hotels, hostels are great for meeting new people and usually have a bar or common space where you can mingle with the other guests. New hostels are being built all the time, and some of them are just as nice as hotels, yet they cost a fraction of the price of a hotel room.
Hardcore backpackers usually do not pre-book hostel stays because they have one precious thing that students abroad do not: time. Rather than waste time searching when you get to your destination, book a hostel as soon as you have booked your train or flights. Better yet, do some research so that you can arrive prepared for what the hostel does (or does not) provide, know exactly where it is located, and spend your time exploring your destination.