If you searched on this forum, then that should not be surprising, as this isn't a hardware specific discussion forum.
It's actually quite difficult to make that comparison. The G3258 is an interesting processor, and it actually makes for an interesting choice for a light gaming rig. Interestingly, when it comes to single core performance the G3258 actually beats the FX-6300. However, the G3258 only has two cores with no hyperthreading support, whereas the FX-6300 has six cores. This means that when it comes to multi-core performance, and multi-threaded applications in general, the FX-6300 will butcher the G3258. Or in simple English: the G3258 isn't going to cope well when dealing with multi-tasking and programs that use multiple threads (AKA "divide their work" over multiple cores). However, in terms of single-core performance (that is, when an application uses one core/thread only) it's going to beat the FX-6300.
So, while you could probably game fairly well on this (granted you use a dedicated video card), I am unsure whether you could use this for livestreaming. That is, normal CPU-based video encoding is out of the question (for livestreaming, anyway). However, you mentioned that you wanted to use QuickSync, which is indeed a technology this processor supports. This will take a lot of the load off the CPU. However, I do not have any personal experience with QuickSync, so I can't help you there.
What do you want to use your future rig for?