The GTX 970 should work fine in your board and would be an excellent choice. The only concern is if your system has an adequate power supply. 

What power supply do you have in your system, make and model?

so i will buy a new GPU first. Then i will look after good other components. i fact, all i want is to play some latest games in fullHD scale in high resolution, like titanfall 2.

do you recommend the GTX 780 or could i buy a GTX 970,too?

thanks for the answers, because i do not know much bout pc´s and, like i wrote, i won't to optimize just a little.

Cheers 

My vote would be keep the 6300 for now and wait for Zen and Kaby lake to hit the market. That board is not only a budget option but it is also rather old now and you may have compatibility issues with new graphics cards.

An FX 8XXX will not improve that setup much even if it were reliable on that board. Honestly, you would be better served replacing the board with something newer. I still have a similar board in use in one of my older systems but that is using a Phenom II and an old Radeon HD 5850. 

What are your full system specs? We can't really help you decide on an upgrade path if we don't know the full specs. 

If you are looking for simply a gaming performance boost then you could look into getting something like a GTX 780 or R9 290 for that board. Those GPUs should work without compatibility issues and provide very decent performance at 1080p. That would net you far better performance than say an FX 8370 + GTX 750ti or AMD equivalent. Put your budget into a new, faster GPU rather than upgrade CPU + GPU with a weak GPU.

i try to follow your thoughts an arguments, nut all in all it is still not easy for me.

as i understand, i could buy a 

for that board. Is that right?

I ask, because I already own that bord with a 6300 Processor. I want to keep that board and change CPU-Processor and buy a new GPU. This, I think, is the cheapest way to optimize that system a little bit.

or are there better way to optimize in the same budget?

I never get hurt feelings when someone speaks from superior experience 

What board would you recommend Wardog? I only mentioned the Extreme 3 because, as you mentioned, it was a cost compromise but still based on the 990FX chipset. I don't own an FX CPU so can only speak from what I have read and other users experiences. My Phenom II x4 955 works a treat in my budget 960GM /U3S3 FX but I have never tried an FX CPU in the board, mostly for the reasons Wardog mentioned, the power design/VRM phases are simply inadequate for reliable long term use. Fewer power phases = lower efficiency + more heat. Add to that the lack of any kind of VRM cooling and you are asking for burned out VRMs and possibly a damaged CPU. 

Simple answers:

Will that board support an FX 8 core CPU?

Yes

Will you be happy with it?

This isn't because it is a bad board, it is simply a budget board. The FX line of CPUs has proven power hungry as well as being brutal on the VRMs of budget and even mid range boards. I will now leave this to Wardog as he is by far more up to date with AMD CPUs than I.

The 8350 is a refreshed (no significant difference over other 83XX CPUs other than clock speeds) CPU that comes with AMD's newer Wraith cooler. 

My advice would be to either go with an FX 6XXX CPU or get a more capable motherboard like this: 

http://www.asrock.com/mb/AMD/990FX%20Extreme3/ (copy and paste the link)

I hope, some of you can help me with the following question:

can you tell me, if the following processor will work with the 960GM-VGS3 FX: 

is there maybe a better or latest processor working with this motherboard?

Micky