Woes of DPS
One of Chris's first problems is obvious: He's a mage. (Cue all you warlocks making jokes.) In the vast sea of players looking for raids and dungeons, DPS is the vast majority. Like, whoah. It's so rare that people can't find DPS to fill out a raid that when a lack does happen, the instance becomes a joke, screenshots are shared to sites like Reddit, and casual bystanders lean back and go "whoah."
While it sounds as if Chris wants to stick with his mage, all you rookies or casuals hoping to PUG into raids should note the problem. You have much, much better chances of getting invited to a PUG or a guild group if you're a tank or healer. Keep a DPS spec in your toolbox, of course, but tank or healer is the way to go.
When a PUG leader is building a raid, he can usually afford to be incredibly selective when it comes to DPS characters. And that leader must be selective. Usually, a PUG is forced to accept any tank or healer it can get, because tanks and healers are in short supply. If a tank or healer is willing to raid, chances are some enterprising guild has snapped them up already.
In order to make sure the PUG has a high chance of success, the leader is forced to pick only the absolute best DPS to fill out the raid. (He might even need to overcome some shortcomings in the tank and healer department.) The leader will look for as many ringers as he can get -- people who have demonstrated their skill and can put out huge DPS numbers.
It sucks to be DPS in this case, but it's the reality of the situation.
Take me as I am
Chris's question boils down to "How can I get a group without changing classes?" He actually used the phrase "take me as I am."
Thing is, if you're trying to get a group without picking up additional gear or achievements, which represents the literal "as I am," your only option is to keep trying and hope to get lucky. Try to talk up the PUG leaders you see, make friends, and network. Hopefully, these folks will see your name in a friendly way, over and over. Eventually, you'll get drafted and rock on to success.
If we're not going for the literal "take me as I am" route, then I would suggest taking a bit of time and going to get the achievements you're missing. T11 content is easy now on normal mode. Guild groups do it to help bootstrap new players into current content, and PUGs can still get decent weapons and such from the end bosses.
You'll have a better chance getting a PUG for one of those groups and then moving on to Firelands success afterward.
It sounds like Chris is probably already rocking pretty good gear. For those of you trying to get into successful raids who want to know how to catch up the curve, the path is fairly simple.
Do normal dungeons first (presumably, using the random Dungeon Finder). When the random Dungeon Finder says you're suitable for non-Troll heroics, start doing those. Keep going. Lastly, when the random Dungeon Finder says you're ready for Troll heroics, go kill some Trolls. (Who doesn't love killing Trolls?)
Try and squeeze in Blackwing Descent, Throne of the Four Winds, and Bastion of Twilight around the time you can start doing Troll heroics. This will help bootstrap you to the Firelands level quickly.
Obviously, doing the 4.2 dailies will help a lot, also. This is just a rough guide, and you should keep looking for PUGs that will take you all along the way.
Most important tip
The most important tip to finding PUGs with a good chance of success is to keep trying. Talk to the raid leaders, and if they turn you down, ask them to consider you for the next round. Keep trying to PUG on your server; keep trying to find new groups. Keep improving your own stats as you go, and eventually, the magic will happen.
All of this will be made a little easier when the Raid Finder tool is released in 4.3. While you'll still need to PUG to get into normal difficulty, at least you'll be able to start with the Raid Finder.
Visit the WoW Rookie Guide for links to everything you need to get started as a new player, from how to control your character and camera angles when you're just starting out, to learning how to tank, getting up to speed for heroics and even how to win Tol Barad.