Savage makes excellent guns that are accurate and easily capable of taking out targets at long ranges. You can get them cheap, or go for higher end rifles. Here are the calibers I suggest, and you can search specific calibers on Savage's website.
The 22 Hornet is an awesome round and shoots over 4000 fps. Look at Savage's bolt actions. It is a varmint round and has a lot of power. I have seen videos of hogs being taken with it.
The .223 is a good round to find in a bolt action, which savage makes. It is capable of 1000 yards with pinpoint accuracy, and is easily capable of taking down a coyote.
The 22 WMR, or 22 Magnum is a good round, but I wouldn't shoot with it over 400 yards, but savage makes bolt action 22 WMRs for as low as $200!! They go up higher in price, but the cheap ones are very accurate especially with a decent scope. While capable of taking down a coyote, shot placement is key because it lacks power compared to the other cartridges I named.
The 22-250 is a good varmint round and Savage makes them at decent prices, and is a very accurate round.
Overall of you are on a budget, definitely go for the .22 Magnum because the rifle is cheap, accurate and the ammo is cheap. Hope I helped and happy hunting!