I currently use Cisco VPN client on Windows Vista to connect to my school. However, whenever I use it all of my connections get dropped; e.g., Firefox downloads, torrents, etc. Is there a way to set it up so that it doesn't interrupt my current connections, or is there another VPN software that does it?
Not unless you control the server.
The server will tell the client that it is to be the default gateway, and that the client is to not let this be changed while the connection is active, and the client will obey these instructions. You can configure the VPN server to allow clients to have more flexible routing, but only if you have admin access to the VPN server(s).
One way around this that I used to make accessing a similar VPN more convenient was to run the VPN client in a VM, that way only the VM was cut off from the rest of the local network when the VPN was active. This way the VM console still works fine so the VM can be interacted with, though without LAN access the only way to transfer information between the machines is text via clipboard integration or by USB drive (by connecting the drive to the VM and host alternately). Depending on what you are using the VPN for, this may or may not be a practical solution.