Dealing with the time between queues
We'll tackle the space between dungeons first. If you're not playing a tank or healer, you'll have at least a few minutes to kill between each dungeon run. This is just the reality of class demographics; the playerbase isn't evenly divided on a 1:1:3 ration (that's one tank, one healer, and three DPS characters, see.)
The time between dungeons needs to be spent farming the vital materials to level your professions. While some cloth, ore, and materials can drop inside dungeons, you're not going to get nearly enough to keep your professions on level. It's a bummer, but it's true.
Hit up sites like Wowhead so that you know where your best farming grounds are. Then, while you're waiting for the dungeon queue to pop, go to town.
New quest paradigm and why it doesn't matter
Once upon a time, in the sad days of yore and 5,000 visits to the same dungeon over and over, you picked up dungeon quests out in the wild world. You'd talk to Joe Bob Dwarf who'd send you into Uldaman to kill a particular rock creature. Then, returning with the shiny bauble, you'd only then discover Joe Bob's close friend Cletus the troll needs to you go back into Uldaman. Rinse and repeat a bajillion times until the mere thought of Uldaman would send you into apoplexy.
Nowadays, all the relevant quests for each dungeon are nicely housed right by the door. It saves an immense amount of time and gives you a nice little XP boost for running the dungeon. That placement is a huge quality of life improvement, but it also doesn't really matter to someone leveling primarily through dungeons.
The XP is a nice bonus, but it's only a small percentage of the experience you'll gain killing monsters and bosses. I bring this up now because most tanks run dungeons like their pants are on fire and only the final boss in a dungeon has the ability to cool those flaming trousers. Trust me, I know; I'm part of the burning breeches crowd myself. While we'd like to believe most groups are cool with taking the time to complete every quest, prepare yourself now to not get everything done.
For that matter, leveling through dungeons is so fast that you might not even get the opportunity to turn them all in. If you don't make it back to the front of a dungeon, you could be ineligible after another random run. So try and do quests, but don't sweat it.
Keep your gear in good shape
The most important thing you can do between each random dungeon is to drop by a local repair guy and fix up your gear. Don't let yourself be the hunter who randomly becomes a melee class because his gun just broke -- or, even worse, the paladin who goes shieldless mid-boss fight.
Choose a group play spec
While most of the specs in WoW are somewhat viable in any playstyle theater, let's be honest. Some talents don't really help out the group; they're best suited to PvP or to solo play. Those talents are cool. They're fun. But they don't really belong in a spec intended to run the Dungeon Finder over and over. This is just basic logic; you're spending all of your time in groups, so choose specs that support groups.
Take the time to help others
As I often do, I ran a bunch of Dungeon Finder groups to make sure I knew of what I speak before I wrote this article. More than any other thing, what made me wince over and over was the number of groups that would ignore the pleas to do a certain boss. Plenty of dungeons have optional bosses and side quests that aren't strictly worth it on a time-to-XP ratio but result in completing a quest.
A new character walking into a dungeon for the first time hasn't completed all those quests. If you can find it in your busy agenda of racing to the end, take the time to knock out those quests for the new guy. Think of the karma!
A special word for tanks
Hey, tanks. Join me over here to the side for a second. You DPS and healers can stay over there. It's just us meatshields for a second.
Now that we're alone, I have a special note for you guys: Put a skull on what you're killing. All those DPSers with the guns and the Lightning Bolts and the Fireballs are just dying to stab and mutilate the bad guys. And, believe it or not, most of them will do their best to be on your target ... if they know what the hell you're targeting. Show them what to kill with a nice, friendly floating skull.
Look, it's just us tanks. I know you switch targets all the time, I know keeping threat can be a pain in the lower levels. I'm perfectly aware that, sometimes, you no sooner target an enemy before some mage opens up with a Pyroblast of Infinite Hells and Screw Your Aggro. And, I know, our pants are on fire and the delicate bits are getting burned.
But, seriously. Put a skull over what you want them to kill. They'll thank you for it and think you're the most amazing tank to ever tank.
Now that we're all back together, this is the single most important thing to remember about random dungeons. Random dungeons are random. And while I've not yet encountered a lower-level group that's not capable of completing a dungeon, the odds are that someone has to pull the short straw and fail to kill Hogger.
If you happen to draw that short straw a half-dozen times, it sucks to be you. But hold out hope that soon enough, you'll be flying through the levels with ease. Leveling through dungeons is fast, easy, and a whole lot of fun. (Also, easier on the feet; you don't have to run around everywhere.)
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.