It'll be virtually the same as if you had your own private server. In other words, you will be able to install pretty much anything you want, and therefore connect anyway you want. But yes, you'll likely need to go in via SSH first in order to set up everything. If you aren't a big fan of the command line, the first thing you set up might be something like Webmin.

Oh, I'm all for CLI. But it seems a bit expensive that I need a whole private server...sounds a bit expensive as well, seeing as how normal private servers are pretty expensive. All I want to do is do web activities through there instead of through the network here...I guess there's really no alternative to VPS in terms of this. How else would I go about it (unless I had my own computer/connection set up somewhere else).

But what would be the difference between getting a VPS and a private server?

Most VPS providers I've seen charge between $8.00 and $20.00 a month.

The usual alternative is to VPN to your house, but like you said, it isn't possible in this case. You basically just need a box outside the college. Pretty much any box will do as long as the Internet connection on that box doesn't suck.

For starters, the price. If you think VPS is expensive then don't even look at private servers. Also, with a private server you get to enjoy all of the hardware's power/capacity, whereas on a VPS you get a slice of it. But honestly a dedicated server would be complete overkill for what you are looking to do. A cheap VPS would work just fine, and it would let you add lots of functionality beyond VPN too.

So if I get a VPS, how would I go about using it? Do I just point all my applications like my web browser and instant messenger to the VPS like a proxy? Or some other way? I've never had any kind of server before.

On the VPS, it could be as easy as installing/configuring OpenVPN. While on your PC, it could be as easy as installing/configuring NetworkManager (with the OpenVPN plugin). You won't need to configure your applications individually.

Since I can't log into the router and check, is it possible that I would be restricted from VPNing to the server by the router if the admin configured it that way?

Possible, but unlikely, for reasons explained in the Networking section of the Wikipedia link I gave you. This kind of thing is easy to test, though, so you can know whether or not it's gonna work before you make any investment.

Okay, thanks for all the help.

Time to go VPS shopping!