Dinging level 60 isn't nearly as fun as it used to be. You're probably already in Outlands, smashing faces with Tier 2-equivalent gear and looking forward to your flying mount. But wait! You've still got one thing left to do in Azeroth. Yep, it's time to head back through the Dark Portal and get your epic charger.
The Alliance and the Horde have vastly different questlines for their epic chargers, and I've got to say, the Horde one is tons easier. The Alliance one is so time-consuming that it might just be easier to shell out the gold for a regular horse/ram/elephantything. But for those who really want the status symbol, The Light and How to Swing It Presents:
Getting your epic charger (Alliance)
First off: if, as you're approaching 60, you can get your hands on Runecloth, Arthas's Tears, Arcanite (actually, start transmuting your own Arcanite a week before doing the quest if you're an alchemist), Stratholme Holy Water, an Azerothian Diamond or a Pristine Black Diamond, hang onto it. You'll need it later.
Your local paladin trainer will tell you that it's time to go see Lord Grayson Shadowbreaker in the Stormwind Cathedral District. Shadowbreaker will give you two quests: Emphasis on Sacrifice is supposed to prove your loyalty to the Light, and The Work of Grimand Elmore will help you get your horse's "barding", whatever that is. Emphasis on Sacrifice is easier, so let's go do that one first.
For Emphasis on Sacrifice, you must bring 150g to High Priest Rohan, a priest trainer in the Mystic Ward of Ironforge. In return, he'll give you his Exorcism Censer, which you can take back to Shadowbreaker. Shadowbreaker will then give you the quest Exorcising Terrordale. Head to Terrordale, the city just west of Plaguewood in Eastern Plaguelands. You need to use the Exorcism Censer on green circles that you'll find on the ground. Be prepared to face 1-3 spirits every time you use the Censer. After 25 spirits, you can return to Shadowbreaker and start work on Grimand Elmore.