While the.22LR and 5.56 have the same diameter, that's the only similarity. The 5. 56 has about 3x's the velocity of the.22lr. more recoil, and much louder report.As others have stated, most predators will be raccoons, but the OP post mentions concern for larger animals like coyotes. You are unlikely to get in pistol range of either unless they are either rabid or trapped.
So you want one firearm that can be used to dispatch both. I'd suggest either a 20 gauge shotgun or an AR15 with a CMMG conversion bolt.
A 20 gauge shotgun (as WingItRanch suggested) will have low recoil, aiming isn't hard due to shot spread, and you can change shot size to make it suitable for different targets. Price is cheap, you can buy a brand new single shot 20 gauge for $150, a pump-action for ~$250, or a double barrel for ~$450. Buying used it will be even cheaper.
AR15 is an option. Chambered in .223/5.56, it will effectively take down a coyote but will be overkill for smaller animals like raccoons. However, the bullet diameter of 5.56 and 22lr is the same and CMMG makes a conversion bolt that allows you to swap between 5.56 and .22lr in just a few seconds. Price varies, you can buy a completed upper and completed lower separately and have a rifle for ~$350 or less, and the conversion bolt is an additional $200 new.
Since you "never liked guns" I'd recommend a simple 20 gauge single shot with a couple boxes of 00 buck and slugs. It will be the cheapest option and easiest to maintain and clear. Practice safely, with a knowledgeable friend or instructor if you can. If you practice at home make sure you have a suitable backstop.
A .22 rifle will be the least expensive to purchase, and to shoot compared to the 20g. Again, .22lr has much less recoil and report then the 20g. For coons, possum, I wouldn't want to use bird shot, unless really close, and I wouldn't want to use buckshot to to the mess it would make to the critter. Wouldn't shoot a coon on my coop with buckshot as I'm only going to damage my coop do to number of projectiles buckshot tosses out, compared to the single projectile of the .22, which I'd gladly use on a critter that is on my coop.
Is a .22 great for fox and coyote, no. But it will kill them. I've killed coons, opossum, fox Coyote out to 60+ yrds with a .22. not doing that with a 20g shooting buckshot. Slugs, absolutely, but loud and much more expensive to shoot.