You can do the 120gr Nosler Partitions at 3,300 fps or the 80gr Barnes TTSX at 3,900 fps for varmints it's pretty versatile. The 120gr Partition can take any game in the lower 48 including Elk without much fuss.
One caliber to cover all of those is tough if you are looking at saving the pelts. I'd recommend the .243 as well. If you were to get a caliber seperate for the coyote then I'd go with the 22-250. I have the .220 swift and although it is an excellent cartridge, the ammo is a lot more expensive than the 22-250 and the performance is not that much better. As Don will probably state as well, the 6.5x55 is also a round to consider.
I agree with any rifle in the 6mm cataglory up to the 25-06 and if you hand load that would be a big pluss. You can have hotter loads for the larger game and then some lower ones for when you want to save the petls. Either that or you just need to learn how to shoot them varmets in the ears.
I shoot a .257 weatherby mark v fibermark I use for deer and antelope. I like the weatherby mark v accumark or sako varmint in 22-250 for coyote or rockchucks. If I had it my way I would have one of every caliber however the wallet does not allow for that but good luck with what you choose
I believe the 243 to be a great coyote cartridge that will work very well on deer and antelope. The 25-06 is a great deer and antelope cartridge that will work well on coyote's. Any other 24 cal and 25 cal will certainly work as well. The 243 and 25-06 will be easiest to find ammo for and be most inexpensive. I have both and have had several of each. My favorite cartridge all around is what CG called, 6.5x55. It still has good varmit bullet's but you'll need to handload or use game bullet's. Same class would be the 260 Rem, I think it is the most un-noticed new cartridge ever and one of the best ever, particularly if you hand load.
For me there's a fine line between varmits and then game up to deer and then game up to elk/moose. I would never recommend any 223 cal for up to deer althought they will certainly kill them. The line between varmit/ big game and big game varmit is at the 243 cal on the small side being better suited to varmit's IMO. And 257 cal on the big side being better suited to deer sized game in my opinion.
So I have right now a 243 thats set up for coyote's, A 25-06 set up for open country deer and antelope, a 6.5-06 set up for long range shooting, and my third 6.5x55 that with a 129gr Hornady will do everything and then some what all the rest will do. Less muzzle blast than the 6.5-06 and the 25-06, less recoil than the 6.5-06, about like the 25-06, better selection of game bullet's than either the 243's or 257's. Which ever one will be best depends on what you have and shoot well and like to shoot! That's probably not the answer you want to hear but that's about the way it is.
A whole new thread could be done on caliber's, not catrtidge's, suitable for all different game. Tremendious overlap of performance today.
First of all, I think everyone has made good choices however I think its not such a thing as a best caliber. I think its all about comfort. I have all types and calibers of rifles but I am most comfortable with my Howa 308, is it my most expensive rifle I own? No Is it my largest caliber rifle? No but it is the rifle/caliber that I shoot the best. So my advice for you is to find a good shooting rifle any caliber from 243 to 300 Win Mag for deer that you are comfortable with and stick with it.
As a Taxidermist I look at hundreds of Whitetail Bucks every year. I know that I have seen many different ways people judge the size of racks on live deer. I think for the most part they all are pretty close. I'm going to give you some measurements that I have noticed being right on in just about every deer. We have used it in the last couple years and have only been off by about 5-6 inches. We do study the deer while bowhunting so we get a little more time to look at them....