Best for YOU hunting coyotes? 223 hands down. PLETHORA (huge understatement) of ammunition specifically for varmints (of which all will work when used). Pretty much if it isn't FMJ ammo, use it on ol wiley. Its also cheaper, easier to find, lower recoil, less muzzle flash, and a flat trajectory for a new shooter (much better than 22lr).
My first "coyote" gun was a Stevens in 223. Bought it cheap in 2009 for that purpose. Now that range is becoming an issue, more or less me being afraid a 55 gr doesn't have the energy past 250-300 yards in windy Kansas, and not to mention the wind correction; I am switching that same rifle to a 22-250. It took me this long to truly master that rifle. Now, for something with a bit more oomph behind it.
A Stevens rifle can be found used for cheap. Or a new/used Axis. No matter if you buy an out of this world custom or a cheap Stevens with a rusty receiver for 50 bucks, don't do what I or many others do: Buy a nice scope, no matter the brand.
Seriously, so many well made and accurate rifles are out there used and new, buying a nice scope will be what you need. Its hard making due with a 24x scope that has ridiculous parallax. You want to be able to SEE your target and make an accurate shot. Its very annoying having to see a tiny spot and follow it for 3-4 minutes.