In addition to whats been posted here to already, here are my thoughts.
First, maybe the best advice on shotguning I've ever gotten- "Take your time....but quickly" It also applies to rifles. Alot of time when we shoot we fall into the "good enough" syndrome and miss when if we hold off just a fraction of a second we connect.
Second, the best shooters I know with few exceptions are those that shoot the most. My advise is is to buy an accurate .22 with high powered optics find a barnyard where you can SAFELY shoot sparrows. Leave the scope on max power (18 or more) all the time and shoot from field positions. Trying to find a sparrow at 40-60 yards can be frustrating at first but if forces good target acquisition. Only, shoot sparrows in the head as its usually very mobile and hard to get a bead on. During this time, I also recomend mixing in a lot of bench shooting for groups. This serves as a constant reminder than the rifle is up to the task which is important when shooting in the field. You know the rifle will hit that bird everytime if you do your part. When you can hit a sparrow in the head with 9 out of 10 shots at 60 yards from field postions you are doing good.
Part two is breaking out the shotgun and shooting skeet or sporting clays once a week. Believe it or not, the mount, swing, shoot of clays will incoorperate itself into your rifle shooting at moving critters.
Shoot 5000 plus rounds between your .22 split between the bench and field shooting, and 2000 plus rounds through your shotgun on the skeet range over the course of a summer and you will put alot more fur on the stretcher come fall. The trick is to practice enough that shooting becomes instinct so that when mister coyote shows up you just shoot it without thinking about the shot.
I made a very difficult shot on the biggest deer of my life using this method. I had a split second to get one shot off and I can remember everything about that shot. It was like watching a telivision. Everything was instinct, I didn't think about the lead, the swing, or trigger pull, I din't think man this is a hard shot. I just pulled up and let the gun shoot the deer and there was never a doubt in my mind where the slug would hit. I wouldn't have made or even taken that shot if I had not shoot at 6 nights a week after work the summer before.
Last tip, if you have access to them, prarrie dogs are your friend.
Matt