Basic Mounts
If you still love your old gryphon or wind rider and simply wish it were slightly fancier, purchase an armored version from the vendor found next to the flying skill trainer. Both gryphons and wind riders are available with red, yellow, green or purple armor. If you want something even fancier, Mei Francis, in Dalaran, will sell an armored snowy gryphon to members of the Alliance and an armored blue wind rider to members of the Horde.
Reputation
If you want something completely new, consider going for a reputation mount. These come in all types, from compact nether rays to majestic drakes, proto-drakes and hippogryphs. Choosing one is much like picking out a car model and color -- except for the fact that, once you've made your choice, you have to convince the dealership to sell it to you. Impress your chosen allies by completing quests, collecting items or defeating their ancestral enemies and you will soon be flying around the world on a fancy new ride.
Achievements
Slow and steady wins the race, or so goes the saying -- and in the case of mounts rewarded by achievements, it is actually true. Far from being instant gratification, obtaining these mounts requires continuous effort over an extended period of time. Complete specific tasks during dungeon and raid encounters, complete a full set of special holiday achievements over an entire year or gather a lot of different mounts to get your shiny new mount delivered straight to your mailbox.
Dungeons
Looking for a challenge? Try for a dungeon mount. These mounts are in the hands, or occasionally claws, of certain bosses in specific dungeons in the game. Defeat the boss enough times and you may manage to obtain one, although the drop chance is very low so it's all about persistence and luck. The mount that is arguably the most coveted in the game, Ashes of Al'ar, belongs to this category, being an extremely rare drop from Kael'thas Sunstrider in the Tempest Keep raid dungeon.
Professions
Characters with certain professions have the luxury of special mounts available only to them. If you are a tailor, you can be traveling around on a blue or red flying carpet straight out of "Arabian Nights." If you are an engineer, on the other hand, you can craft yourself a turbo-charged flying machine, complete with flame decals and a dancing hula-girl doll above the control console. If you are an alchemist, or you have a friend who is one, the vial of the sands will teach you how to transform into a drake and carry an additional passenger on your back.