One must hold a spring-piston air rifle ("springer") more loosely than a conventional firearm. Consistency is extremely important. High velocity is stressed as a sales ploy. The last thing you want to do is approach or exceed the speed of sound with a pellet (in my neck of the woods, around 1100 fps) . . . accuracy goes plumb to hell at that point. Sample pellets, find one your particular piece likes, and stick with it. Never clean the bore "squeeky-clean" or excessively, either; accuracy will fall off. Never use conventional solvents; use a high quality airgun oil (I use the Beeman product); seals will suffer otherwise.
Remember follow-through; with a typical springer, the pellet is still in the barrel while you are feeling the recoil. Use a "soft" rest as opposed to sand bags . . . a rolled-up towel works fine & rest near the balance point of your rifle (typically an inch or two ahead of the trigger guard). If your scope mounts don't feature a scope stop, get some that do; without this feature, scopes can & will move around on air rifles. Check mount fasteners periodically.
Off a dead rest, I can maintain 1-ragged-hole groups with my Baikal IZH46 pneumatic target pistol (10 yards), < 1/2" groups with my HW97 Beeman, and always < 1/2", typically much less with a Beeman R-7, if I do my part (the last 2 at 25 yards). As far as pests are concerned, I "own" my yard (in the world of Starlings, I believe the silver maple next to my shop is widely referred to as "The Tree of Death."). I use nothing but Crosman Lights in the Izzy & R7; Crosman heavies in the 97.
Good luck . . .