First, you'll need an IRC client for your operating system; Lunarnet's preferred picks are listed in the navigation bar to your right. Once set up, simply point it to irc.lunarnet.org at port 6667 and 4444, and you'll be good to go. This, however, will only bring you to the IRC server itself; from there, you'll need these commands.
Basic IRC Commands
/server - This will connect you to a specific server. If you've already set up your IRC client to connect to irc.lunarnet.org, that should get you onto the network. However, if you're having problems you can try connecting to a specific server via, say, /server eldrasia.lunarnet.org.
/join - This will make you enter a channel on the network; a chatroom, in effect. Generally you'll be joining a specific channel but it's worth noting that you can join #[channelname] and if it doesn't already exist, it'll create it in case you want to tug several people in there to discuss something.
/part - The opposite of join; makes you leave a channel. You can leave a message after the channel name, e.g. /part #[channelname] Time to sleep.
/quit - Does exactly what it says on the tin; quit the server. Similar to /part: /quit Time to sleep.
/me - This will make you emote; that is, perform an action in the 3rd person e.g. /me writes an IRC tutorial -> * [Nickname] writes an IRC tutorial.
/nick - Changes your nickname (screen name, IRC name, what-have-you.) /nick [Nickname2] -> *** [Nickname] is now known as [Nickname2]
/msg or /query - Use these two to send private messages to someone, e.g. /msg [Nickname] Hi. for example. The difference between them is that /msg will display the sent notification in the channel you're sitting in, while /query will open a separate window for you to chat to that person.
/topic - Changes the topic of the channel. Generally speaking only people with % or @'s by their name can do this.
/ping - If you want to test if you're still connected to the server, here's an easy way: /ping [nickname] should hopefully return a line something like [Nickname PING reply]: 2 secs. What this means is that it took 2 seconds for your ping to get to the other user and their reply to get back to you, and because it did you're still connected. You may instead /ping #[channelname], but this is sometimes frowned upon.
/whois - Look up information on a user, such as their hostmask, what channels they are in, how long they've been inactive e.g. /whois [Nickname].
Servers and Staff
Getting Started in IRC
mIRC - Windows Client
Colloquy - Mac Client
BitchX - *nix Client