The input side is a LED. From datasheet fig.3 its Vf will be about 1.2V @10mA. 5mA If is the *MIMIMUM* to guarantee a new MOC3063M will turn on at 25 deg C, but the LED optical output will fall as it ages + it requires more current at lower temperatures, so it would be prudent to design for double that - 10mA. If you are stuck for drive current capability, you can squeeze the margin a bit but I wouldn't go below 6mA.
As the input side is a LED, it needs a current limiting resistor. Assuming 5V rail to rail logic levels, the resistor must drop 3.8V, and pass 10mA. 390R is probably near enough, though you should also allow for the voltage drop in the MCU's output pin driver, so 330R or even 270R may be a better choice - check your MCU's data sheet for output voltage vs load current.
It will turn on whenever the input side LED is on *AND* the voltage across MT1, MT2 is less than the inhibit voltage (typically 12V). This effectively prevents turn-on before the voltage zero crossing. It then latches on until the current through is output side falls below 500uA - that's the current zero crossing, which will only be the same as the voltage zero crossing if the load is purely resistive. If the LED is off, it will then turn off.
I wouldn't recommend totally omitting the snubber as even with a resistive load, wiring inductance can be enough to be problematic. With an ordinary TRIAC, you'll probably need a low capacitance snubber network. For a so-called snubberless TRIAC, I'd lay out a snubber network, then leave it unpopulated so its possible to retro-fit it. See
ST's AN437 for snubber network design.