Here are my thoughts on places in Europe that I've been to:
London - you have to come here. End of story! If you're staying a few days you should look into taking a day trip or 2 to places like Oxford (which I'm kind of biased about but it's gorgeous) or Bath.
From London you can get the Eurostar to Paris or Brussels which is quick and fun. I think if you're young (which you are!) and book ahead it's not too expensive, but I'm not 100% sure. You can also fly - which I tend to do because there's an airport 20 minutes away from my house that has really cheap flights. (and by really cheap I mean sometimes they're £1 plus tax). For cheap flights to/from/within the UK try
www.easyjet.com and
www.ryanair.com - the airports aren't as close to the middle of London and sometimes the airports the other end aren't exactly in the places you're going to (especially with ryanair) - but the flights are really cheap and I know in Brussels they laid on a coach to take us from the airport to the city centre. There are lots of other cheap airlines around. I used to have a list, I'll see if I can find it.
Paris - I <3 and you should definitely go there. Lots to see - museums, old cathedrals, general sightseeing and shopping.
Brussels - has a lovely old square, but other than that I didn't like it that much (apart from the parts that involved OLP) - lots of dodgy foreign men trying to talk to you if you're female - but you shouldn't have that problem. Lots of fish restaurants!
Amsterdam - great. Again easyjet will have really cheap flights from London. It's good just to wander around, lots of cobbled streets and big squares with cafes. Also the famous red light district! And more serious things like the Van Gogh museum, and Anne Frank's house.
Berlin - I haven't been for ages, but I loved it. I think things will have changed a lot since it was only a couple of years after the re-unification when I went.
Zurich - is really nice, up in the mountains. I don't think we went into the city that much, as we were staying in a youth hostel further down Lake Zurich (I think)
If you want to stay in youth hostels there's a pretty big network across Europe and you should be able to book online. Depending on how much you want to spend, it's probably worth looking at
www.expedia.co.uk or
www.opodo.co.uk or lastminute.com as they often have cheap deals.
As for travelling between places - flights can be very cheap as I've said before. The train network's also a lot more extensive than in north america and you can get a pass for all of europe or sections of it (at least I know you used to be able to - it's been a while since I looked!).
STA travel whose website that you found are the main student orientated travel agency over here so should be useful..
I think you need to spend at least 2 weeks over here.