While I typically agree with finding what is most available locally, .243 Winchester ammo is pretty widely available online at most larger retailers and can be ordered/shipped to anyone legally able to purchase from anywhere in the country. Same can be said for the rifle, as well, by having it shipped to your local FFL, or your local big box store may be able to have one shipped to them from another location.
Since this is a first rifle, and coyotes are mentioned, a .243 is a great choice as it is a good caliber for coyotes (regardless of user experience) and the cartridge itself is pretty soft in the recoil department- something new shooters often appreciate. The 6.5 Creedmoor is also a great cartridge, with a wide availability of factory ammo, but will have a little more recoil than the .243 Winchester. All of this can be reduced on either rifle with a nice muzzle brake, though. We can assume distances will be relatively short (under 300 yards) at first, and with either the .243 or the 6.5 Creedmoor, you can grow into longer distances as you practice and get better reading wind.
For a light, handy rifle around the homestead, look for a smaller profile barrel with a shorter length (22" would be common on a factory rifle). Stainless barrel/action with a composite/polymer stock will hold up against the elements, but a classic blued finish with a walnut stock will pull at the heartstrings of even the most hardened individual. And this will be your rifle, so you want to love it and you want it to fit your personality as well as your intended use.
Go to as many stores as you have time to visit, and try as many rifles as you can (even if they are not in a cartridge you think you'd be interested in)...holding them will let you see how it feels in your hands/against your shoulder. Cycle the bolt, pull the trigger if they'll let you. All of the little details will impact your final decision, and you have every right to be as thorough as you want.
Good luck!