The stories of fast leveling could be due to the player already having played RoM to higher levels or endgame, and therefore access to more gold, twinked gear, or just knowing the best quests to take and ignoring others. (For example, getting a 1 million XP orb and 1kk Talent Orb from the AH would get you to lvl 30 instantly). Fast leveling could also depend on class, since some classes are not built to do damage (ex. knight) which means it take longer to complete quests while others can kill multiple targets and complete dailies in no time (ex. mage). Learning the important skills, stats, and strategies of your class is also important. (ex don't stack magic attack on a melee/physical attack class).
Occasionally there are 'Double XP' weekends that double xp for quests, but not for tp which can gimp your skills in the long-run. "Double XP" events happen randomly, but sometimes close to holidays. RoM's 3rd anniversary is coming up soon so... maybe then.
If you are starting out, leveling could take some time, especially if you haven't played an MMO before. But if you want to get to endgame faster, you could always take the highest level quests because they offer the most XP. This could mean getting to The Coast of Opportunity as that is intended to level a character's third class and so gives higher xp.
You could just focus on leveling only one class, and then using high level dailies to level your secondary class. (i.e. you can turn in quests up to 9 levels higher than your current level, so you could do quests as a lvl 30 mage and then switch classes to your lvl 21 rogue to turn them in)
Grabbing all the available quests and checking to see which quests require the same kills can also speed things up. (ex sometimes killing the same boar can net you 3-4 quest completions.)
Getting in with a good guild can be an option too, as some can help you blaze through the quest-lines and instance quests. Teaming up with friends can also help you finish a whole zone in several hours.
Also, sell everything you find and buy the best weapon possible from the AH. This will help you kill monsters faster to finish the quests. It could also help you fight higher level monsters to get more xp and tp. If you want to spend money, you could trade for gold to buy the xp/tp orbs, or buy xp/tp bonus potions.
You could study
http://www.runesdatabase.com/ to figure out what the best quests are.
I'd agree not to level
too fast as you may miss game content, class skills, tactics, or other aspects of the game that become important later. Save power-leveling for an alt on your second time through.