|What StuartS said.|
I will tell you this much, VLAN's aren't necessary to this assignment.
A VPN is completely the wrong technology. A VPN is used when you wish to go from one private network to another and have to traverse a public network (like the internet). What you're talking about is going to be pretty much contained within one room (data center) so no VPN is required.
Here's the last bit of help I'm going to give you. You need each server to be on two networks at the same time. One, the network that allows clients to access resources on the servers. The second, for your private network so the servers can communicate with each other.
Come back when you're done and show us what you came up with and I'll be happy to offer some input.
There is no one single "correct" answer. I can think of two different ways to do this off the top of my head and both are equally as correct.
It matters not how straight the gate,
How charged with punishments the scroll,
I am the master of my fate;
I am the captain of my soul.