Late May/early june is right at that transition from the "shoulder" season to "high" season, when prices for everything go up and the crowds roll in. If I were going to Europe and planned any time in big cities, I'd probably try to go in May rather than June. In terms of duration, 2-3 weeks is good, particularly if you're backpacking/hosteling. After awhile, you kinda want your own space again.
I would echo the comment about STA (student travel agency). Nearly all major universities have an office. They can get great discounts on airfare and train passes. That's another thing altogether; whether or not to get a Eurail pass. Depending on where you plan to be, there are regional passes and continent passes with unlimited rides and some combination of # of rides in so many days. Look carefully at what you're buying. If you're itinerary is not totally set, you may find that that availability of regional passes will guide which countries you want to go to.
If, on the other hand, you're going with 3-4 people, renting a car can make a lot more sense. But, again, this depends on your itinerary. If you're going to spend all your time in major cities, take the train, even if the car would've been cheaper split 4 ways. You don't want to bother with parking, plus, with the train, you can often get overnight trains between major cities which saves you on hotel AND you avoid wasting a day in transit.
Anyway, there're a million guidebooks which will cover this kind of stuff better than I can, but these are some prelimenary things to have in the back of your mind as you get started.
Whatever you decide, do SOMETHING. You're unlikely to have this big a chunk of time totally uncommitted ever again.
