Stop by your faction's tent and speak with your reps.
The first thing you'll need to do once you've picked up the quest is to grab a lance, without which you can't ride the mount to which you've been directed. The lances are located in weapon racks located all around the Tournament grounds, but the two most convenient racks are right in your faction's tent.
Just right-click on the rack to receive your Lance. And make sure to equip it! If you don't, you can't mount your trusty steed.
Speaking of, you can't hop on any mount just yet. There's a specific Stabled Quel'dorei Steed or Stabled Sunreaver Hawkstrider that has your name on it. Find the stables, located on the southern side of the Tournament Grounds, make sure your Lance is equipped, and right-click on the Steed.
Once mounted, you receive a new set of controls, like most vehicle mounts.
Each corresponds to an action you need to perform on the target dummies, located around the Tournament Grounds. You can enter the Valiant's Ring and engage any of the dummies there or find some of the scattered dummies -- any labeled dummy will grant you quest objective credit so long as it's attacked in the proper fashion.
Tournament Ranged Target: Hit your Shield-breaker button at range to break the target's defenses. Do this twice.
Tournament Charge Target: Engage the dummy by pressing the Shield-breaker button at range and then hit your Charge button while its defenses are down. Do this twice.
Tournament Melee Target: This is a little more tricky in that the dummy will counterattack when you Thrust. To prevent yourself from getting two-shot, make sure to have a full stack of Defend (3 stacks for 90% damage reduction) before you rush in and use Thrust five times.
Once you've completed your target practice, return to your faction reps -- and revel in the superiority that comes with experience. Maybe soon you can ride a mount you don't look ridiculous on!
Reward: 13g23s at level 80 and 2 x [Aspirant's Seal].