Here is where I want to go:
Austria
France (Paris and the beach)
Germany
Greece
Italy
Netherlands
Luxembourg
Slovenia
Spain (Barcelona)
Sweden
Switzerland
I know I can go to all of these places in a week, so where should I leave out? Also, should I have one trip on spring break, and then take smaller trips (like Germany or Sweden) on the weekends?
Don't waste money on a Eurail Pass because you will not get value for money. Work out exactly where you want to go, and buy point-to-point tickets - it will be much cheaper.
The distances between the countries you want to visit are vast - it's not a one hour trip, you need to allow a full day to travel between countries. Look at seeing just two cities in two countries. At weekends visit countries that border Denmark as that is all you will have time to see.
Have a look at a map and look at the distances between Copenhagen and the places you want to visit, and check the train timetables. For instance, Copenhagen to Paris takes 13 to 14 hours one way! And you will need to change trains twice. The beach is nowhere near Paris. Presumably you mean the Mediterranean in the South of France, well, that's another six hours travel! If you want to see beaches to the North of Paris, that's around two hours. And you will need to add in connection times for these trips.
You can't visit more than two places in a week and get anything out of the experience. One place would be better. Also check the train schedules as I think you will be surprised at how long it will take you to get to the Mediterranean coast from Copenhagen.
An obvious option is Paris, a week in the city will > begin < to give you a feel for it, and you can realistically go to Paris by train from Copenhagen. No need to buy a Eurail pass if you go to Paris as you can just buy a round-trip ticket.
If you want to go to a beach then look at Croatia, such as between Split and Dubrovnik. You have to fly as you only have a week off from school.
Check the typical weather for your destination(s) before you decide as you don't say when is spring break for your school. For example, Switzerland has absolutely miserable weather during April.
If you're in Denmark and you're a student, you'll be broke. The closest affordable places are Poland, Czech Republic, Slovakia and Hungary. I'd go by train to Bremen, Germany and fly to Budapest direct on Ryanair. I did this six years ago. I bet you can fly there for less than $90 USD.
One week? Forget euraipass, not worth it. Also choose one country (say Germany and opt for perhaps something like 3 days in Berlin and 2 days in Dresden). Fly to Germany or use Deutsche Bahn.
You cannot possibly visit all those places, I would stick to Central Europe. Slovakia, the Czech Republic, maybe Austria.
i can give you some nice destinations between france and north west italy ...you can make a good travel visiting... lyon annecy chambery (in france) and aosta torino (in italy) you have to run but...