If you are over level 50 then I recommend doing the Cape Jule or Pelladane race, this is because they start and finish right next to portals, so very little unnecessary travelling.
When you pick up the quest you will be teleported to the start circle and a countdown will start, you need to roughly follow the race track path hitting all the white checkpoints til you reach the end. The checkpoints are marked on your map with a small yellow quest dot. You can go off the track to avoid the red circles. If you do accidentally hit a red then you can use the speed buff (button 1) to get rid of the debuff.

White Circles – Checkpoints that you have to tick off as you go through the race track.
Green Circles – Give you a charge of Endurance which you can use to boost your speed by pushing the ‘Terminal Velocity’ button on your temp ability bar.
Red Circles – Applies a debuff to you that slows you down. Try to go around them!

If you stand still for too long then you will be kicked off the mount and fail.
You have 2 mins 40 seconds to finish.
At the end of the race there is an NPC to hand in the quest to.