After the Cataclysm, the Barrens of Kalimdor were ripped apart and divided into two new regions – the Northern Barrens, and the Southern Barrens. With the North and South Barrens split by a chasm full of lava, it's a little bit trickier to make your way south than it was pre-Cataclysm, but it can easily be done if you know the right path to take.
On foot or mount, you can cross into the Southern Barrens is through the southwestern-most section of the Northern Barrens. On your map you will see the very edge of the lava flow, and there is no chasm in the way. You will need to keep an eye out for Alliance NPCs nearby, but if you are careful you will easily make your way to The High Road and reach the Hunter's Hill flight path & quest hub in the Southern Barrens.
The simplest way to get into the Southern Barrens is to just fly there. If you are at least level 30, the Hunter's Hill flight path should automatically be available to you, as well as the rest of the flight paths. This is a lot faster than walking or riding on a land mount!