Your current Athlon X4 860K only fits on FM2+ socket motherboards which only supports DDR3 RAM.
All Ryzen motherboards are AM4 sockets and DDR4 RAM. If you are getting the R5 1500X, then motherboard choices would be either the B350 or X370 chipsets (supports overclocking). DDR4 RAM supported is from 2133MHz to 2400MHz to 2666MHz. You can fill up all RAM slots in your motherboard (2 slots on certain mobos and 4 slots on most other mobos). Number of RAM sticks (and GB capacity) is dependent on the motherboard selected. RAM speeds higher than 2666MHz RAM speeds is limited to only 1 RAM per DIMM channel (cannot use all four RAM slots on a 4-RAM slot motherboard).
For Intel, assuming you are getting the current generation (Kaby Lake) i7 or i5, are LGA 1151 sockets and also only supports DDR4 RAM. Motherboards can be any of the B250, H270, or Z270 chipset - depending on the features and number of ports/connections you need, as well as the CPU selected. Only the Z270 supports overclocking which is best paired with a "K"-version of the i7 or i5 CPU you will choose. DDR4 RAM speeds for B250 and H270 mobos are limited to 2133MHz and 2400MHz. Mobos with Z270 chipsets can support higher speeds (2666MHz and up).
All of the above CPUs, MBs, and RAMs mentioned can support your current EVGA GTX 1060 3GB ACX 2.0.
If this PC is for pure gaming purposes only, I'd recommend getting an i5 (for its 4-Core / 4-Thread), as few games (if any at all) utilize more than those number of cores/threads. The i7 has hyperthreading which makes its 4 cores run at 8 threads. Same goes for the Ryzen 1500X (4-Core/8-Thread).
Best price/performance motherboard for the AMD Ryzen 1500X is any of the B350-chipset motherboards (instead of the more expensive X370). The 1500X is unlocked (i.e., overclockable) while the B350 motherboards supports overclocking. The 1500X also includes a CPU Cooler in the box, so no need to purchase an aftermarket CPU Cooler for this upgrade.
The choice for an Intel CPU (and subsequently, its motherboard) would largely depend if you plan to overclock or not. As I mentioned above, an i5 would suit your needs if the PC is used for gaming only.
I would recommend the i5-7500 or the i5-7600 if you do not intend to overclock (now or in the future). These two CPUs are locked (non-"K" versions), has lower TDP (heat/power draw), and already comes with a stock Intel CPU cooler in the box. Best motherboard to pair with any of these two CPUs would be the B250-chipset motherboard.
If you wish to overclock an Intel CPU, then get the i5-7600K. This is an unlocked (K version) CPU, has higher TDP (heat/power draw), and does not come with a CPU cooler. You would need to buy an aftermarket CPU cooler for this purpose (highly recommended is the Cryorig H7 or Cryorig H5 Universal - affordable but good performance). With an i5-7600K, the motherboard to pair it with is the Z270-chipset motherboard.
In all of the aforementioned motherboards, you would also need to determine the size (ATX, mATX, or ITX). As you did not mention if you will use your current chassis, make sure it fits the motherboard size you will select.
Budget also helps so we can suggest good parts depending on how much you are willing to spend.
Hope the above info would help you.