Hi,
First of all I think it's great that you're taking this trip. I took a similar trip a year and a half ago (college graduation present) and it is perhaps the single best thing I've ever done. I had so much fun, and at the same time got a real and genuine appreciation for how things work over there. I was gone six weeks, but I saw a hell of a lot during that time.
I'd be more than happy to share my advice with you, but you need to decide what cities you want to see in each of those countries, and how you want to divide up your time. If you have no idea, first thing you should do (do it regardless) is pick up a guide book. The best are made by Lonely Planet or Let's Go (mainly geared for younger people rather than old fogies

). You will take this book with you on your trip too, but it can be a very helpful planning guide as well.
I assume you're in North America, so you will be flying into one of those countries. I suggest London, England or Paris France, just because you're probably likely to get the cheapest fares that way... hopefully you want to visit both of those cities (I highly recommend both). So whatever city you fly into, start your trip there and spend however many days. The countries you want to visit are pretty far apart (Sweden versus Italy, for instance) so you may want to look at whether flying to some of them would be a better option.
If you're going to take more than a few train trips, then it's usually worthwhile to get a rail pass. You can check out the various options at
Rail Connection. You'll just need to sit down in front of a map of Europe and map out the best route to see all of those countries.
Another site you should definitely have is
EuroTrip. As well as normal site content, they have a message board with a wealth of good info. Using the search function there, I was able to get a lot of great info I wouldn't have found otherwise.
I hope you're planning to backpack and stay in hostels, because as well as the cheapest and easiest option, it's also the best way to go when you're young. I wouldn't have done it any other way. About hostels though, they vary in how nice they are. You will find good hostel recommendations at Eurotrip and in the guidebook you will pick up shortly (right?). You're going during the summer, which is good weather-wise, but a couple of problems are that Europe's tourism is at its busiest during the summer months, and also, many Europeans get time off in the summer and so some shops may be closed. But I went in August and early September and was okay for the most part. One piece of advice which I hope you will take (learn from my mistake) is to book hostel reservations at least a few days before you arrive in a city (if not all at once beforehand). When we got to Paris (our first stop after London), we got in at 18:00 (better learn your military time--that's all they use over there

) and started frantically calling hostels... every single one in our guidebook was full up. We had to stay in a hotel the first night, and then were able to get accomodations for a hostel for the rest of our stay. As if once was not enough... had the same problem in Venice. Unfortunately, Venice doesn't have many hostels... so even though I arrived early enough in the day, they were still all full. I ended up staying in a hotel room for three days there.
So, bottom line is plan, plan, plan. Deviating from your plans once you get there is one thing (I got drunk with some new friends and ended up accidentally missing my train from Munich to Berlin... had to go a day later... haha), but at least you have your plan in place if you need it. If you need to change your hostel reservations, most places will do that no problem with a day's notice.
Like I said, get a nice, comfortable backpack, and a smaller daypack, and that's it. Pack them well, but don't overpack. I guarantee you will take more than you need, but try to pare it down as much as possible. Your shoulders will thank you later.
I'd be happy to answer any other questions here or via PM or whatever. Just ask away. Of the countries you listed, I've only been to two of them (England [London and York] and Italy [Venice, Rome, and Florence]) so I won't be much help on what you should do and see in the other countries, but that's where EuroTrip is invaluable. But I learned a lot about Europe travel in general, and I'd be happy to help you further. You're making me jealous... I want to go back!
l2c