those notwithstanding, I must add the following:
Am new to this thread, but have been shooting spring guns since 1975 and air guns since, well, dirt was invented.
I will take no less then 1000 shots to ge the rifle "grooved" and broken in. In that time, experiment with pellets, as has been noted.
The "fliers" are usually from "dieseling" and accompanied by a louder than usual crack, followed by the smell of burned lube. This will settle down in time, but is not unusual even after 1000.
The last thing to consider is that a springer needs more to be cradled than held tight into shoulder like a firearm. If you rest the rifle, it is better to rest it on the stock, and maybe even with your hand beneath...as the recoil power cycle demands a slightly eased approach. (If gun is a gas ram, the shooting is more stright forward, as there is not the same cycle-much faster lock time also.)
You should also review airgun forum.net for many more detailed stuff from very serious shooters from UK and Spain, etc. While they mostly shoot limited power guns (non FAC) due to restrictive laws, they do hunt and shoot paper with a relish generally reserved for firearms in the USA, so heed the information from the ones who shoot spring and gas ram rifles.
Fun, no matter what.