Many players have chimed in so far with their fondly remembered misadventures. Tirion Fordring's Redemption chain out of the Eastern Plaguelands gets many recommendations. As does the Marauders of Darrowshire chain in which you have to find a little girl ghost's missing dolly and along the way relive a critical moment in the history of the lands.
You can also summon a helicopter out of Azshara, get killed on purpose out of Un'goro and learn the history of the Moonkin which takes you to the same area where you can quest for a mechanical yeti. But there is one quest that is getting the most enthusiastic recommendations.
The quest of the Scepter of the Shifting Sands is two very long chains that stand out because they include, as one player puts it, "the chickensuit, draconic for dummies, funny little side bosses like Dr. Weavil, number 2, Azuregos' dialogue, arcanite buoys and so forth." The quest is used to open the gates to Ahn'Qiraj, but it can be undertaken even after the gates have been opened on your server. One warning about this quest: it requires access to the raid instance Blackwing Lair and to be able to kill one of its denizens.
What exciting, interesting or just plain fun quest chains would you recommend to a player before jumping into the Outlands?