25-06 is a very good round. My brother-in-law hunts elk with is as well.
I would recommend two things however.
1) Use a bullet with rapid expansion. I'll never forget the goat I shot with my 7mm RM using 150gr core lokt's. Shot it three times and he kept walking. Finally dropped. I thought I had missed or my barrel was bent. When I gutted him out, you could cover all three shots with a tea cup! No damage, all went right through!

2) Practice long range shooting. 200 to 400 yards. Know your balistics (drop and windage for whatever round you choose). I see you're in IL and probably don't need to shoot at those distances. Goats can be under a hundred, or out to 400. And I can just about guarantee it will be windy.My longest shot was 385 yards. Dropped like a wet card table.

I love antelope! Right next to my mashed potatoes.