I want to create a simple LAN conference-chat style messenger in Java but I have no clue where to start.
It must have the following features:
Though your question is pretty vague you seem to have the basics (sockets and all that) in place. I suggest you start by reading the All About Sockets and All About Datagrams Java lessons on Oracle's site to get started. The main content from the second lesson you may want to digest is the part about broadcasting (for the purposes of automatic server detection).
Here's how i'd go about the implementation on a high level:
There obviously are numerous ways to make this kind of an application. I'm not saying that the way that i described is the best. It is, however, probably suitable to the use case you described, and its implementation is fairly simple.