Full list of compatible CPUs with your motherboard: https://www.asus.com/Motherboards/A58ME/HelpDesk_CPU/
And no, FX-6300 is not compatible because it requires an AM3+ motherboard whereas yours is an FM2+.
As for the best option for your motherboard, it would depend on your current system. If you have a discrete graphics card then there's the case for the x4 880k (I would favour the newer x4 845). If you don't have a discrete graphics card then the best APU would be the A10-7890K.