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You also may have a mage friend offer to port you to Shattrath once you are of level for Outland. Though there's no harm in doing so, the quest progression for Outland starts at the Dark Portal so it is suggested that you first enter through there.
Getting to the Dark Portal
This was formerly difficult, but Patch 3.2 trivialized it. Every city has a portal to the Blasted Lands, where the Dark Portal is located. These portals are found near the mage trainers (if you don't know where the mage trainers are, ask a guard). You must be level 58 or higher to use the portal to the Blasted Lands.
When you arrive in Outland, you will automatically have a flight path to Honor Hold, which will be your main quest hub for early Hellfire Peninula. In Honor Hold, you can find quests for Hellfire Ramparts which is part of Hellfire Citadel, and is the first instance in Outland.
Getting to Shattrath City
There is one Capital city in Outland, Shattrath, which is a neutral city open to both Horde and Alliance.
To get there, you can either cross Hellfire directly west to Zangarmarsh and then follow the road south to Terokkar (you cannot miss the city when going this route, it's very obvious), or you can take the southern exit from Hellfire into Terokkar Forest, and then turning west when the road does.
There are two Inns in Shattrath, one on Scryer's Tier and one on Aldor Rise. On your first visit you can set you hearthstone at whichever you wish as you will be neutral with both, but once you choose either the Scryers Faction or the Aldor Faction (the other will not like you anymore), you should set your hearthstone to the matching inn.
Returning to Azeroth
The best way to return to Azeroth is to use the Major City Portals located in the center of Shattrath City