Pretty much everything you need is in the two games. The Terrans have been fighting the Vasudans for 14 years, battling across a network of subspace nodes that connect different stars. Then the Shivans start turning up and fighting both.
For my money the best way to experience the FreeSpace story is to boot up FreeSpace 2, read the techroom intelligence database to get the story of FreeSpace 1, and then play FreeSpace 2. But a lot of people would advocate playing both games!