Cheap WoW WOTLK Classic GoldBehind the bank in Darnassus is a pink pavilion that's a portal to the base of Teldrassil, a place call Rut'theran Village. There are three docks there. The one to the west has a ferry to Azuremyst Isle, the Draenei starting zone and the Exodar. The southern dock has a boat to Stormwind harbor.
The eastern dock is out of service. There is also a flightmaster there, and you can get a flight to either the Exodar or to Lor'danel in Darkshore. Also, in the Temple of the Moon in Darnassus is a portal to the Exodar.
Anyway, as you've probably noticed, the old world was revamped significantly in Cataclysm. It wasn't just the terrain, but the quests as well. A character starting in Kalimdor will almost never be sent to the Eastern Kingdoms for anything, until they're ready to go to Outland at level 58. That's not to say you can't go; you can and if you're leveling in BoA gear you probably should at some point.
From Darnassus the easiest way to Stormwind is to exit the city via the portal at the west side of the city near the (new) flight master, then take the boat at the south dock.
Later on, you may find use for the other boats: One runs from Ratchet to Booty Bay (be careful, Ratchet is still in enemy territory, though the goblins there don't mind you coming in to town; if you have Friendly reputation with them then they will roflstomp anyone who tries to gank you). Another runs from Menethil Harbor in Wetlands (a short flight from Ironforge) to Theramore Isle in Dustwallow Marsh.
It USED to be that the Darnassus boat (technically the Auberdine boat, but) would take you to Menethil Harbor in the Wetlands and you then had to RUN to Ironforge. We typically referred to this as the "Newbie Crocolisk Gauntlet". Getting your face eaten by a croc ten levels higher than you was a right of passage, dammit.
Nowadays, yeah, just hit up that tree with the glowing roots behind the bank (the one that takes you to the flightmaster) and then head for the boat dock. One of the boats - the dockmasters will tell you which - will take you to Stormwind. If you want to get to Ironforge, just take the tram from the Dwarven District once you're in Stormwind.