Under most circumstances the answer would be...... any centerfire you can hit with... but all hinges around if your planning on saving fur and maybe the specific "varmint" your wanting to hunt.
If you want to get into most calibers "considered" varmint rounds..... common would be 17 Rem, 204 , 222, 223, 22-250, 243 and in some circles...25-06. Then theres a few custom calibers reloaders like such as the 220 swift, 221 fireball...ect.
If your just planning on a gun / caliber for predator control ..not saving furs.... any centerfire which you may deer hunt with will do. Also lets you get a taste of varmint hunting without spending extra coin. Yotes specifically.... can be pretty challenging. Been a bunch of people invest a lot with high expectations ..dont always work out that way.... then they give up all together. Which...nothing in the world wrong with hunting yotes with a 30-30, 30-06, 270...ect. If you can hit with it...your good to go.
If your planning on skinning & saving furs.... the 223 and under are probably more favorable....and with the AR popularity ... the 223 is probably the most economical to shoot. However critters such as bobcats..... the 222, 204 & 17 Rem are known cat getters without blowing big holes.... but yet very capable on yotes.
From the 22-250 on up.... they can make a big hole but definitely do the trick. Which in my biased opinion.... I use a 22-250 on yotes and overall it's probably the most common & favored caliber by the majority. Ballistically its hard to beat , scary accurate and definite will reach out a very long way.
As for an "all-around" hunting caliber..... hard to beat a 243. One gun / caliber you can use for varmints and deer. Low recoil... accurate....and can buck the wind much better on long distance.
On the very low end of centerfire but your really need to consider keeping shots closer.... 22 hornet...and the new 17 Hornet...which new as in its been around for awhile (17 K Hornet) but normally a custom reload round but now Hornady is offering factory ammo and believe Savage is offering the gun.
Rimfires..... most would say the minimum is the 22 mag for yote sized critters and again keeping shots close( 75 yds & under).... but I've had good luck with the 17HMR using 20gr CCI Gamepoints and restricting distance as much as possible. Prarie dogs, ground hogs, ...small vermin in general.... their not a bad choice at all....and the 17HMR is scary accurate as well.
Also..... would have to include a shotgun in the mix. A 12 guage... right choke & allowance to use buckshot now.... makes a good close range yote getter...especially in timber.
The gun (manufacturer / model) .... thats a Ford / Chevy / Dodge debate. I personally like Savage / Stevens for models... but thats me...and all depends on what you want & how deep you want to get in your wallet.
Hope this helps.