There is a very straightforward tutorial on how to set up your own Symfony project: http://symfony.com/download
I will not copy the tutorial, just mention two points:
To install and upgrade Symfony, you will have to use Composer. (Details in the tutorial.)
Symfony code is actually hosted on Github, but those are only some of the components. The Symfony framework is a collection of libraries, some developed by the Symfony project itself, and some developed by third party projects (Doctrine probably being the most notable).
Have fun using Symfony. It may not be easy in the beginning, but you’ll love it after a couple of weeks. Be sure to read the Book and the Cookbook thoroughly, this will dramatically increase the fun.