I would be mostly just trying to become a better marksman.
A couple of other thoughts:
1. Consider buying a very accurate .22 in addition to a centerfire rifle, such as an anschutz. This will allow you do do a lot of "practice" shooting and really hone your skills.
2. Do you reload? Probably the best way to get the most accuracy from your rifle.
3. There are some great resources out there that can be very helpful. Bryan Litz has a couple books and a DVD set available. I have only read 'Applied Ballistics for Long Range Shooting' and found it full of useful information. I felt it provided a very solid base of information for precision long range shooting. Looking forward to reading/watching his other works. I purchased the DVD, 'Accuracy for the Precision Rifle' released by Magpul, but features Todd Hodnett. I think you would find this DVD very helpful and interesting as it covers many topics from the basics of how to hold a rifle (only shows prone), deal with wind, choose a scope and stock options (he really advocates the AX stock for tactical situations), and a whole lot more.
4. Many have had good luck with trued Remington 700 actions, but there are probably better ways to go, something like a Stiller Predator action with a nice barrel and McMillan stock would give you a dream rifle. Throw in a Jewell trigger and whatever type of bottom metal you prefer and you are probably at about 70% the price of the AX308. The Predator action is based on the Remington 700, but held to much tighter tolerances to optimize accuracy. There are also many other very good custom actions out there.
5. Do not skimp on good optics and mounts. They are link between what you see and where your bullet goes. If your scope can't hold zero or is moving in its mounts, you will never achieve acceptable results.
6. Work on the basics and hone your skills. There is no substitute for practice and experience. (see #1)
7. Most shooters at ranges are more than willing to answer your questions and help you in any way they can. The shooting community is pretty cool in that way. Most will even let you shoot their rifles, look through their scopes, etc. So, respectfully ask questions if you have access to public shooting ranges. Again, no forum response or magazine article can show your eye what a scope looks like or how a stock feels.
Good Luck, would be interested in knowing what you choose.