Almost all motherboard manufacturers publish a list of compatible CPU's for their motherboards. So I googled yours and went to Gigabytes website and here is their list (click on CPU Support tab):
https://www.gigabyte.com/us/Motherboard/GA-F2A88XM-D3H-rev-30#support-cpu
In general, the higher number processor is more powerful, i.e. A10-8750 will be more powerful than your A8-5600K.
So, Now you may want to head over to PassMark.com and search for the processor you have and the processor(s) you are considering. This will give you a score that you can compare. Higher number is again better, but it is most useful to figure out the amount of performance gain, for example, if a processor has a score of 5000 and you are looking at one that has a score of 10,000 you can expect a really good increase in performance.
So, your processor scored 4339:
https://www.cpubenchmark.net/cpu.php?cpu=AMD+A8-5600K+APU&id=1449
I have a rule that I don't upgrade unless I can get something with at least 50% increase in score, so you should be looking around 7500 or higher scoring processor.
And here is the bad news, the most powerful processor on that list is the A10-7890K which gets a score of 5749, which is only around 32% increase. For me that isn't enough to justify the upgrade but that will be up to you.
You may want to look at other things to upgrade, like going from mechanical hard drive (HDD) to a solid state drive (SSD) which is the best bang for your buck as far as performance upgrades go.