All raids can be found either by looking at the Calendar or Rapid Raid sections of the website. Before you can signup though, you'll need to claim your characters on the Guild Roster. Click this link to see an image showing you how to do this.
If you can't see your character listed, contact someone with Raid Leader, Guild Leader or Guild Master rank and they will be able to either update the roster or advise you when this can take place. The Armory usually takes a few hours to update and show that you've joined the guild (and manual updates can only be done once every 16 hours), so it may take anything from 5 minutes to a day for your character to show on the roster.
Once you have claimed your character, you can start signing up for raids immediately. To do so, click on the Rapid Raid link above and click the signup button next to the raid you wish to sign up for. For an example, click this link. This will take you to the signup page for the raid in question, and on the right you'll see three buttons under the heading Signup: Available, Tentative and Unavailable. Available means you will be there on time and ready for the raid invites going out at 19:30, Tentative means you think you will be present or will be present for only part of the raid and Unavailable is pretty self explanatory. Click this link to see an example.
When you select one of the options, you'll be presented with this pop up window:
As stated, you select the character that you plan to raid with and enter any extra info that may be of use to the person doing the invites, such as if you will be late, need to leave early, or if you happen to like pie and then click submit to continue.
Once you've hit submit, that's it - you're signed up for the raid. To see if you've been picked for the team, check back on the day of the raid and go into the Rapid Raid section. Hover your mouse over the the (17/17) bit next to the signup link and you can see the team that's been picked. For an example, click this link. You'll notice there's a Waiting List section at the bottom: This is for people that we'd like to be available "on the bench" for the raid. Whilst on the Waiting List, you'll receive DKP exactly as if you were in the raid, and may be asked to come in at any time. This may be only for a single fight, or it may be to replace someone - be prepared either way.
You can update your signup status at any time up until one hour before the raid.