Learn the apprentice riding skill which is now obtainable at level 20. Go to your respective mount trainer and pay the price. of 4 gold If you're a Warlock, Worgen or Paladin then you will complete the achievement when you hit level 20:

The best way for a new toon to get it's very first mount is to practice your professions. Especially mining, skinning, or herbalism. Then sell all of your materials on the auction house.

Horde Mount Trainers

• Hawkstrider: Speak with Perascamin in Thuron's Livery, Eversong Woods.
• Kodo: Speak with Kar Stormsinger in Bloodhoof Village, Mulgore.
• Raptor: Speak with Xar'Ti in Sen'jin Village, Durotar.
• Skeletal Horse: Speak with Velma Warnam in Brill, Tirisfal Glades.
• Wolf: Speak with Kildar in the Valley of Honor in Orgrimmar.

Alliance Mount Trainers

• Horse: Speak with Randal Hunter in Eastvale Logging Camp, Elwynn Forest.
• Saber: Speak with Jartsam in Cenarion Enclave inside Darnassus.
• Ram: Speak with Ultham Ironhorn in Amberstill Ranch, Dun Morogh.
• Mechanostrider: Speak with Binjy Featherwhistle in Steelgrill's Depot, Dun Morogh.
• Elekk: Speak with Aalun at the entrace of The Exodar.