Outland travel guide (Alliance)
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 leading straight to the Blasted Lands right next to the Dark Portal. These portals are found near wherever the mage trainers are; ask a guard if you aren't sure where. You must be level 58 or higher to use them, however.
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
In the Terrace of Light there are permanent portals leading to each of the four Alliance Capital cities (and each of the four Horde Capital cities too, unusable by the Alliance). Many find it helpful to set their Hearthstone to Shattrath so they can get to any city in Azeroth they wish to quickly.
There are also portals to Stormwind and Orgrimmar right beyond the portal in corresponding camps.