Every time I turn on my VPN for the last couple of months, I suddenly can't access the internet at all anymore - Why?

Well, we need a little more information to help you with this one. Are you trying to VPN to your home network remotely? Your work VPN from home? You home VPN while at home?
When you establish a VPN connection all of you traffic is usually being tunneled into the remote location. This can cause a lot of lag and make it feel like the internet is not working. A VPN is primarily used to gain access to files and services remotely, and is not always set up properly for browsing the internet wile connected.

I’ll tell you what I know. I have cable internet and a wireless network at my house, and my school has a network with resources (dictionaries, search engines, course webpages, etc) that typically person cannot access unless they are connected to the school network.

My VPN allows me to access the school’s network from my house. The point of giving students VPN is to allow us access to the school’s internet resources from home. When I first got it, I could access these websites. It’s not that it’s slow, I can’t go to a single website when the VPN is on now, not even google.

Great, that totally clears things up. Just to check, you can’t access anything while connected over the VPN? Not even the schools resources? If this is so something is wrong with the connection. I would contact the school’s tech support and make sure that your computer is still set up properly.
There is a chance that something is wrong on the schools end, or they made changes to their firewall and you need to re-configure your VPN client.