I'm going to stick with the building your own path. This should be a lot of fun and very educational as a personal project, but if you're trying to make it for work or for real users, follow Bart's path to something already built.
Anyhow, forget servlets. This will be much simpler than that! You can start a Plain Old Java Object from a main() and open a ServerSocket and then you're in business. The
Sun Custom Networking Tutorial is quite good. Work your way through that, code and run the examples, and you'll learn a lot of what you need.
And for grins, here's part of a server I made.
You should be able to explain what it does pretty well by the end of the tutorial.