Congratulations on your new gun. As you can already tell you are going to get some people who's first responce is "get a bigger gun". Nevermind them, you have a great hunting gun. Yes you will need to get a bigger gun to consistantly kill coyotes and you will probably have some coyotes run off. But here is my recomendations to limit the losses and get you on the right track.
1) Don't trust the site-in from BassPro. Site it in yourself, they might be close but close won't cut it.
2) The Bushnell scope might give you troubles. Not that it's a bad scope but it is designed to shoot at targets 200 yards away, not 50 yards. If you can swing it, get yourself a Simmons 22 Mag scope. Great rimfire scope and is set up for the ranges you will be shooting. I have had one on my 10/22 for 4 years and it has NEVER lost point of aim. Check Ebay. Some great deals on there and if you are patient you will be able to pick one up real cheap (between $10 and $20)
3) Practice makes perfect. Shoot as often as posible. I recommend every other night at first if you can get out that much. Shoot at least 20 shots but no more than 50/per day. And except for sighting in, never shoot it from a "shooting bench" or other pre-arranged shooting support. Shoot it from all sorts of positions, sitting, kneeling, prone (lying down), etc. And remember, a critter is not going to show up right in front. Practice shooting from odd angles in those positions. Example: You are sitting facing south and the target is to your left in the east. Shoot a 1000 shots thru the gun or until you are able to hit a 2" ( 1" better) target at 50 yards from all sorts of odd angles, 9 outta 10 times! Make it a game with yourself to see what kinda contorted positions you can put yourself in and still hit your target.
4) Use CCI Mini-Mags. I have yet to see a gun that didn't shoot this round well and it should be more accurate than the CCI Stingers. With small calibers such as the .22 cal, accuracy is going to be the most important item. The extra punch you get out of a Stinger won't do you much good if it doesn't make a good hit. A good hit with a Mini-Mag will kill just a quick as a good hit with a Stinger. And Mini-Mags are cheaper to boot.
5) As it said in the movie, "Aim small, Miss Small". Don't just aim at the head. Aim at the eye, or better yet the corner of the eye. The smaller the target the better. Ignore everything else, just concentrate on your small target.
6) Limit your shots to 50 yards until you can drop a yote at that range everytime.
7) Hunt, Hunt, Hunt!!! Hunt everything your 10/22 can legally kill. Coyote, fox, ground squirels, jack rabbits, rats, raccoons, etc.
Keep us informed and have a great time!!
Good Luck,
Chris