If you don't own a .223 caliber rifle yet that is what I would buy first. The ammo is everywhere and easier to find in many bullet styles and weights. The 22-250 would be my second choice since it is the classic varmint rifle caliber. It is fun to shoot but ammo is sometimes hard to hard. You will find one box here or there. The 243 would be my last choice since it is on the biggest side for varmints and the smallest side for deer, but non the less a known great caliber. I own two 243 rifles and rarely shoot them unless I'm strictly gonna hunt long range coyotes in heavy wind or small Cali deer. Lately, the rifles I find myself hunting with the most for varmints are my 223 and 222 because I might encounter a Bobcat or Fox and don't want to tear them up.
Thanks to everyone for all the advice. I guess I should have posted some results quite a while ago. My parents ended up getting me a Remington 700 SS in .243 for Fathers Day that year. I have already taken a few coyotes and 3x2 blacktail with it. I guess they made the decision easy for me. The gun fits me well and I absolutely love it. Thanks again!!!