Which libraries/platforms are available on Linux to start game development?
OpenGL and SDL are both open source and cross platform.
For C#, there is also a Mono library, which attempts to be a game library like XNA but cross-platform. I have not used Mono, but heard bad things about it.
PyGame is another popular cross platform library.
I don't know much about Java libraries, but Java itself tends to be easy for simple beginner projects, and it is obviously suited for multiplatform stuff.
The Unreal Engine apparently works on Linux using OpenGL, you might be happy to hear.
In general, anything open source will probably support GNU/Linux. More than that, they usually "start" on Linux. Proprietary stuff rarely strays outside Windows, on the other hand.
Keep in mind that if you develop for Windows, using for instance DirectX 9, your games might be playable through Wine on Linux if you're lucky.