Those critters can make for some long shots so I use a .300 mag or a .30-06 by Remington (700 BDL). But a .270 will do fine for those who dislike recoil.
Because the "average" shot distances for speed goats,mountain goats and Dahl sheep are farther than "average" shots on deer and elk-and they are smaller animals, you can fudge a little with velocity over mass. The cartridge most recommended is actually the 257 Weatherby Mag. It doesn't have the same recoil as the 300 Weatherby because it uses a modified / shortened 300 H&H case instead of just expanding it. It is hard to beat the 257 Wby mag for shooting flat and true, but possibly a 7mm Mag or 270 WSM would come real close too. But load up a 117 grain Barnes or other top quality bullet and you have a real distance shooter. I'm not normally someone to recommend a Mag,but this is a real exception. And the good news is the Weatherby Vanguards come in 257 Weatherby,and they are having a big promotion event to sell Vanguards as close to $399 as possible. Now the bad news,if you really want to shoot the rifle enough to get proficient,but not so much to shoot out the barrel you might have to hand load-or go bankrupt buying cartridges. If you choose the 257 Wby, you will have a unique and capable rifle. If you went with a different cartridge the 7mm Mag,the 270,30-06 and 300 Win Mag would all be good choices too. And if I were to get a non-proprietary cartridge, I think I'd still stick with the Weatherby.
I hand load, so some cartridges would be great to play with, I just don't have the need for a Weatherby Mag in the arsenal.