The cartridge choice is really almost irrelevant in your case. There is little difference between most hunting cartridges. They all shoot a chunk of lead at a pretty good speed and kill stuff. If you are looking at 500 yards and under then a .308 will do fine on any animal you will shoot. Bigger cartridges from there are only more expensive and recoil harder.
I think you have two options. Go get a nice Tikka rifle and shoot it. Shoot it a lot. Custom barrels won't really help you with learning the art of shooting a high powered rifle. Time behind the gun is the key. Sure most people know that but most don't shoot more than 10 rounds a year through their high powered rifle (AR's excluded).
Option two (this is the option I would choose) Sell all the guns you have and come up with enough money to actually build one bad ass custom rifle. You will be very, very happy and this will help improve your accuracy in the long run. In reality you only need one rifle for everything in the lower 48. If you need a bigger gun it will probably be a be a really big 30 cal or 338. Deal with that later. Guys spend way to much time talking about guns and buying every caliber known to man (you do realize that new cartridges are developed to sell more guns to guys that need every caliber). I'm a firm believer in getting to know one rifle intimately and being very proficient with the firearm. You will be a much better marksman when you limit the number of weapons you hunt with. I've got a Bat action chambered in 6.5X284, brux #3 stainless steel barrel (carbon is way over rated) jewel trigger, Mcmillan stock and topped with a zero compromise scope. It drives tacks and I'm comfortable shooting as far as needed. I'm not cool like all my buddies that have 5 rifles and can say that they have all kinds of options. But.....they know when we go shooting I'll crush them.