Firstly, this is no small undertaking at all. I recommend you possibly take out consultation and if you are in the UK, I am happy to help!
That being said, I would go about it via the following:
Why do people need their own VPS? Do they need internet access and do you need people to be able to reach them from outside the network? This will determine the need for public IP or private IP.
Heavily depending on usage and the amount of students, you need to choose what resources to give. Testing may be 256-512MB of ram and a 20GB hard drive - development or more could be 1-2GB of ram and 60+GB hard drive...
After this, do you want to keep them all on at the same time or a lot of work before people need them and switch on just what is required.
If you want some on at any one time, you may be able to get away with having a lot less equipment and some extra hard drives, it will be work intensive starting and stopping VM's, but, it will save thousands on hardware and not impossible.
If you want everything on at once, again, depending on the amount of students, this could mean a lot of hardware equipment.
In most situations I would create a new IP range for this project, you either want to reserve a few addresses in the subnet for the hypervisor on the machines themselves or a different subnet.
As for the Hypervisor itself, for a pay solution, ESX/VSphere is the superior solution - you could load it on to every machine and use features and load balancing across all the machines available to you. It is by far the most complete solution... but also the most expensive by far.
Next, my personal choice would be Microsoft Hyper-V Server which is free. Hyper-V server is amazing for scripting when combined with Powershell. If you have the hardware and environment all set up, you could set up all the machines with minimal work - that being said, *nix performance on Hyper-V can be problematic (depending on who you speak to).
Honestly, there isn't an easy answer and we need to know a lot more information to give you a serious answer, but, I hope this puts you on the right track.