<Snipped quote by DesperatePerson>
In an RP setting you probably shouldn't get into to much technical details. The only times it would be appropriate to bring up stuff like where you put your hand, how you sit the arrow on the string etc, would be 1: If you have a character discussing it in dialogue for some reason, such training, or 2: If you need to build tension related to the shot.
So the best way writing-wise would be basic. "Jim guessed at the wind and let loose an arrow" or something to that affect. Most people's imaginations will fill in the blanks. One way I heard it explained is that, in writing, you should only describe the stuff people couldn't work out themselves based on the information available. A person using a bow to shoot an arrow is probably going to follow the same process we expect to see in movies, so you only describe it for effect, or if they are doing something weird with the bow.
As for the process of shooting an arrow, it isn't particularly complicated. The arrow has a notch. You place the notch against the string while gently holding the notch in place, then you rest the shaft on the hand holding the bow, pull the string back with the notch-hand, get an idea of where you are aiming, and let loose.
With novel shit like this youtube can really be helpful. I'm probably not the best person for bow advice because I've only played around with them for hunting and fun stuff, and I am not at all a serious archer (The only bow I even own myself is a crossbow). You can however find serious archers, or serious anything, all over youtube. There are several weapons channels that are just dudes nerding out about stuff like this. There is also how-to videos, and videos about experts enjoying their field. Sometimes the research can be more fun then the writing.
This was extremely helpful, thank you!