I've decided to make this topic because a lot of people who connect to IRC do so via webclient, which is fine, but somewhat inefficient if you want to spend a lot of time on IRC. I mean, who wants to have their browser open alltimes, right?
Here I'll attempt to make an easy tutorial for those who have never used an IRC client before.
Due to the fact that I like XChat and it's one of the more simple yet effective clients, I'll be using this for the tutorial. I'm sure then when you get to grips with IRC you'll change clients if you can't stand this one heh.
Windows
If you're on windows you can get the XChat IRC client from
~~ http://silverex.org/download/
Mac
If you're on a Mac you can get the XChat IRC client from
~~ http://www.macupdate.com/app/mac/11070/x-chat-azure
Linux users can obtain XChat in many ways, for ubuntu users you can do this via the Software Centre.
After Installation
Once you have installed XChat and run it you should have something that looks like this (slight variation between OS);

Connecting to Freenode
To connect to freenode, the IRC network all arma related channels are on, you should click on;
XChat --> Network List
When you have done this a box should appear;

Fill in all the information at the top with the nickname you would like to use to connect.
Then, scroll down until you find "Freenode" (As highlighted in the picture) and click "Edit".
At this point another box should appear;

You should leave the section "Servers for Freenode" as is and you can also tick the box named, "Use global User information".
You'll also want to tick the box named, "Auto connect to this network at startup". This will mean that the next time you launch XChat it'll connect to Freenode without you having to manually do it.
Once you have done that you can click the little button with [...] next to the "Favourite Channels" section.
After doing so you should see a box like this;

Once inside you can click "Add" to add all the channels you would like to connect to.
Here is a list of some regularly used Armagetron channels;
#armagetron
#armagetron.pickup
#armagetron.tourneys
#swift-clan
#team.unk
#rogue.tronners
#crazy-tronners
(The # at the start of the channel names is important, don't remove it).
When you have finished adding the channels you would like to connect to, you can press okay, then press close on the Freenode box as well. Now all that is left to do is click the Connect button on the Network list and you should be well on your way to connecting to Freenode.
Extras
Once you have connected to Freenode there are some things you can do to make sure your name doesn't get taken by someone else.
You can register your nickname by registering with NickServ.
For more information simply type;
/msg NickServ help register
and follow the instructions.
I hope this was useful for some of you and happy IRC-ing!