NBK said:
Buy the Tikka T3 in .223 and be done with it. It comes with a great trigger adjustable down to 2 lbs., a 1-12' twist barrel which is perfect for 45-55 grain bullets and it only weighs 6 lbs. They have a sub MOA guarantee out of the box too.
Ah...no brainer!
x1000 on this. We have 3 T3 Lite Stainless in the family, 2 of them in .223. After a very quick zero (like less than 5 shots), my brother shot about a 1.5" group at 200 yards, brand new out of the box.
Besides being a flat shooter, the .223 is going to be FAR cheaper to shoot than the other cartridges you mentioned (.220 swift, .22-250, etc.). It's going to take PRACTICE to hit squirrel-sized targets out past 300, and practice gets expensive.
I should have read the entire thread before replying. It is VERY important that you define "varmint" before deciding. It's really too broad of a term. If you mean rodents (squirrels, prairie dogs, etc.), you can go through (literally) thousands of rounds in a trip, if you get into the right area. Cost becomes a critical consideration, as does recoil.
If you want a dedicated predator rifle, again, you have to practice if you want to hit something past 500 yards. But since you will probably shoot far fewer rounds at predators, something with a higher ballistic coefficient could be warranted. There are plenty of guys shooting 1000 yards with .243's and .260's.
Like John said, the .223 is a pretty good balance. It will put coyotes on the ground, and won't break the bank, especially if you're handloading. In my ARs, I'm shooting 77gr bullets (1:7 twist barrels) at about 2700 FPS, which doesn't go subsonic until it gets past 800 yards. At that distance, it hits about the same energy as a .22 magnum does at the muzzle. That's not spectacular, but if you shot a coyote point blank with a .22 Mag, could you put it down? It all depends on......
Wait for it........
SHOT PLACEMENT!!!!!!!!!