Instructions
Consider the mounts you already know as well as the ones that may be easily available to you. You may already know one or more mounts that you can match with a pet; if not, you will have to purchase one.
Decide which mount you would like to match first with a pet. The easiest mounts to obtain are ground mounts for races belonging to your faction. If your character belongs to the Alliance faction, purchase saber cat mounts from the night elves: for example, you can match a Striped Frostsaber mount by taming a Frostsaber from the Winterspring zone, or a Striped Nightsaber mount by taming Shadowclaw, a rare beast found in the Darkshore zone. If your character belongs to the Horde faction, purchase wolf mounts from the orcs or raptor mounts from the trolls: for example, you can match a Timber Wolf mount by taming Ripp, a beast found in the Hellfire Peninsula zone, or an Emerald Raptor mount by taming a Highland Scytheclaw from the Wetlands zone.
Travel to the location of the beast that matches your mount and tame it to become your pet.