I wouldn't get a mini-14. It is inherently inaccurate due to the reliable but "sloppy" mechanism. I have a nice mini-14 and after spending 6 months working on the rifle and testing various types of ammunition, I managed a 4 inch 5-round group at 100 yards.
By comparison, my 7mm rem mag (browning a-bolt with BOSS accurizing and anti-recoil system) shoots a 1 inch 5-round group at 200 yards.
A 22-250 shoots flat and accurate, and a nice 270 would also work well. My brother picked up a bolt action 30.06 for a few hundred bucks and it shoots ok, but it's a little abusive on the shoulder for casual varminting.
The 7mm I have would explode a rabbit and pretty much mangle a coyote, but with the right ammunition (lightweight with composite tip and boat tail) it would be accurate enough to hit a coyote out to or beyond 500 yards depending on conditions and the quality of the rifle. Use heavy game ammunition and the same rifle can take an elk. I got this rifle because I wanted one rifle that could take anything I had any desire to hunt. I don't want to hunt bear and I have no desire to pack a moose out of the woods even though a well placed shot will take down a moose too, but it's perfect for almost everything else and since there's only a 4-6 inch or so drop at 300 yards, hitting what I aim at isn't much of a problem. If I sight it in at 200 yards, I can shoot out to almost 500 yards and the bullet will be within 6 or so inches of aimpoint, plenty accurate for a kill without much aimpoint guesswork.
One of the 22 magnum rounds would be cheaper and there are tons of varmint hunters buying up accurizing kits so you might be able to find a nice hopped up 22 magnum (hornet?) that would take coyote out to a few hundred yards without trouble. My Dad has a 22-250 and it's quite nice. Imagine a .223 round, but with an additional 1000 fps
For cheap varminting fun, even a ruger 10-22 can be accurized. My 10-22 will shoot 3 inch groups at 100 yards bone stock. Adding a bull barrel and a trigger kit would cut that in half, making it perfect for rabbit and ground squirrel. It'll take a coyote if you hit it in the right spot, but I don't think I'd want to risk a non-fatal shot so I'd use .223 or some other magnum .22 for coyote or larger. .223 is also good for deer if you can place the shots, but again you risk the animal running off and slowly dying somewhere else if your shot isn't placed right, and that's just not cool.
Just don't get a mini-14 and expect to hit anything with it. It would work up close against a coyote and they're reliable and easy to carry, but it's really not a good varminter and I haven't talked to a single gunsmith who said they can be accurized worth a damn.
My brother has a long/heavy barrel variant of the olympic arms AR, (M-16) and it's very accurate for a .223. I'd trade my mini-14 ranch rifle for one except that the olympic arms rifles are very expensive due to the idiotic bans the weapon has faced because it looks scary. My mini-14 cost $350 and so far it's eaten 2 scopes due to the abuse the heavy bolt dishes out; My brother's AR is currently running around $1800 and faces state-specific export/resale restrictions. It's a bear to carry for long distances because of the heavy barrel but it points easily and hits right where you aim. Right after finishing Army boot camp, I saw my brother pick off 2 running jackrabbits at 75 yards with that thing. Just point, shoot, and mr rabbit exploded in a puff of fur and bloody goo, just like in the video games
edit - for the price of the accurized mini-14 shubie mentioned, you can get a nice AR-15 variant. Another drawback of the AR however is that every damn federal agent, whether it's FBI, BLM, or the girl scouts auxiliary, will try to confiscate the damn thing on sight. The mere sight of it invokes instant fear and hatred by the anti gun freaks and the fed jackboots. A stupid BLM guy tried to confiscate 4 weapons from me, my 2 brothers, my dad (while he was an active CHP officer no less), and a friend. This idiot was out in the middle of the california desert when he came across our little group. We had valid id (my dad had his badge too) and valid CA hunting licenses, and every gun was 100% legal, but this jackprettythang tried to take our guns. He must have had a deathwish because if we were "that type", he'd have been buried on the spot and nobody would ever have known where he went. As it was, he knew he was wrong and was lucky to be alive about 5 minutes into the encounter, but he wouldn't back down and called in our names and the serial numbers for an AR sporter, a mini-14, my dad's CHP issued service pistol, and my brother's olympic AR. His office told him to not be stupid, and he let us go. If we hadn't been about the most honest citizens on the planet, that idiot would have disappeared so we still have no idea what possessed him to hassle us. If we were the type of people that deserved law enforcement action, he would have died on the spot. Since we didn't jump him, there could be no possible reason for him to hassle us, but he still did.
So the moral of the story is if you get anything that looks like an M-16 or other assault weapon, expect to get hassled.
