Air Weapon Shooting
First, observe the same safety procedures as a real firearms always keep gun pointed in safe direction, trigger finger must remain outside of the trigger guard until ready to shoot, always unload completely after shooting, use the right kind of ammo, know your target and what's beyond it, etc.
Air rifles don't have much recoil to them springers have a very slight back and front recoil that I don't really notice.
CO2 rifles are affected by temperature, lower velocity when shot in cold conditions). In terms of actually shooting, breathe normally and right before your shot, take a deep breath, exhale about half of it, and then shoot.
Also, find out which eye is your dominant eye / master eye, and use that one for shooting. You may wish to use a benchrest some sort of shooting stand if you find the rifle too heavy.
Also, remember that the pellets are generally better for some things than others (ie. wadcutters are best for shooting paper targets as they make clean holes, domed and hollowpoint are good for hunting, etc. Also, try out different types of pellets, as some guns like some pellets more than others, and that is not to say to match brand of rifle to brand of pellet - I can tell you from experience that for instance, Crosman rifles may not always like Crosman pellets.
Also, if you are shooting indoors, make sure you have a sufficient backstop, so that if you miss or if the pellet goes through your target, it won't go into your wall or something.
Usually aside from properly cushioning behind the target, hanging a heavy blanket behind everything is a good idea. Also, it would be a good idea to wear eye protection (and I don't mean prescription eyeglasses, but protective goggles.
Some CO2 airguns may be loud enough to be heard by adjacent home dwellers, so keep that in mind - usually spring-loaded break barrels are rather quiet.
Observe your target, shut the non-aiming eye, keep the sights upright and put the tip of the foresight on the point of aim and aim in the centre of the aperture. Pause breathing when taking fire, but make sure you're comfortable and not holding your breathe.