Those you run I have a question
Like Day Dream Believer said, a good pair of running shoes is critical to not only running, but running without injury. Because you only have one ankle that hurts, it leads me to believe that its your shoes (one being worn out more than the other). I've never worn compression socks, don't actually think that will help stablize a weak ankle. What WILL help is to strengthen your ankles. Here are two exercises that will help:
Get any kind of balance board/disk/roller, and balance on it one leg at a time. The unsteadyness of the board/disk/roller not only helps you develop a strong core, but the shear fact of balancing on one leg at a time will also increase your ankle's stability.
Sit, cross one leg over the other and write the alphabet with the big toe of the leg that is crossed over (cursive of course). Then switch legs and write the alphabet with the other toe.
Here's a link to at home exercises you can also do to build ankle strength: http://www.wikihow.com/Strengthen-Your-Ankles
Kudos to you on beginning a running program! Get a great pair of running shoes, increase your time running very slowly, increase your distance very slowly, and be sure to stretch completely after each exercise session.