Help me plan my Europe trip!

From personal experience one thing I would recommend above all is to safeguard your passports and personal information when jumping country to country. Europe is not as goody-two-shoes as America and it's pretty wild out there so you have been warned!!

With that being said, I think 2 nights in small cities are good as you would want the most time in the larger areas. If you're studiyng in London then you should definitely take advantage of the locality. You can get to Paris by Train so try to visit that while in London and leave the hard to reach areas for afterward. I see you are going to Italy but only to Florence, I'd take advantage of the train system and visit Venice and Rome while you're at it.

Since you're on a budget you might want to stay in hostels as they are cheap and easy to find. Location matters the most and when you book early you can get a better room at the lowest cost. Also, Europe loves trains so you will do most of your traveling that way, be prepared to shell out around $100 on getting around alone. RyanAir is also a very popular option (and cheap) means of flying around.. its first come first serve seating so get there early and travel light.. they have high fees for luggage.

P.S. what are you trying to do on your trip? Sight See? Party? Shop? Depending on what you are trying to do should influence your direction.

This information is based upon my personal stay in Europe when Studying Abroad as well.

Cut a city or two out.

You don't want to be on a death march, and since this is your first time, you are likely to screw something up - you don't want to be somewhere you immediately fall in love with, and yet have a ongoing feeling of doom hanging over you because you only have 33 more hours there.
 

London is awesome - you don't make clear if your friend is meeting you there, and if they have been there before - they may want to see stuff in London.

I travel with a 2 wheeled duffel bag and a messenger bag large enough to hold my laptop + camera gear as my one carry on. I have had trouble making weight with the checked luggage with Ryanair - I was about 20 days + 5 countries deep into a trip

Hostels are good for the social aspect, and for laundry - I refuse to pay hotel rates for laundry. But traveling in a pair may mean bed and breakfasts are comparable price wise.

Prague is awesome, but it isn't particularly near anywhere else either of you are talking about going to.

I think you have a good variety of cities, since none of the ones you listed are particularly similar to one another.

However, you just picked some of the most expensive cities in Europe to visit (London, Paris, Amsterdam). I am not sure what you meant by cheap, but I studied abroad in Europe and the above 3 were probably the most expensive cities in Europe I visited. Granted, you can see a lot of things in those cities, but it's nice to know beforehand that you will (should) spend more there than in Florence or Barcelona.

If you want to have cheap fun, cities in Spain, Prague, and Berlin are nice. Florence is also a good choice.

I used hostels which, for the majority of them, were safe. I'd probably pick higher rated ones in Barcelona and Florence, mainly because the cheap ones are a bit sketchy and annoying to deal with since they don't have a 'real' front desk (so staff is only on call sometimes).

I recommend easyJet or Ryanair and taking just 1 bag. The train is only convenient if you're moving between London and Paris/Amsterdam/Brussels. Otherwise for your international destinations, take the plane.

In addition, plan your trip ahead of time and book your tickets for the Eurostar (train involving London), easyJet, and Ryanair ahead of time. They are much, much cheaper if you book earlier. I remember tickets for Ryanair were as low as 3 euros for a flight, although it seems like you don't have much time before you start.