couple considerations considering your budget and skill level.
1. Look at the savage target action Fclass, Benchrest and Palma rifles. You can get them in 308 for right at $1000. Another barrel in 300 Win will cost you $350. $60 for nut wrench and go guage and swap barrels in 5 minutes.
Savage Models 12 and 112 Review
good article here on how they work.
A 308 shooting the Berger 185s is a good LR round and you will learn wind doping. Easy to reload Win or lapua cases, either Varget, IMR4064 and RL 15 powders, BR 2 primers and the Berger 185, 178 Amax or 175 SMK. Used Savage target action guns are around $875 to 900. On the 300 Win, same on cases, F210 primers, H4831 powder and either the 190-210 grain bullets from Berger, Hornady or Sierra are the tickets. The savages are great and very accurate guns that are routine winners at Fclass matches.
Fred Moreo of Sharp Shooter Supply normally keeps a couple target actions for sale that are already tuned and tweaked for just over $500, add his trigger and pick your stock and barrel(s).
2. Most military bases are running F class matches and you can shoot the 308 in the F-TR matches. Lot of experience learned fast and easily with guys who will go out of their way to work with you. You will find them on the west coast too. Look here, Benchrest Central and Long Range Target shooting.com for a listing.
3. $300 for bases and rings is Rolls Royce level and not remotely needed for your budget now. The burris double dovetail bases $45 and set of signature rings with inserts $50 will give you the needed 20 MOA. They are as durable as any $300 set up and do not require lapping in. Just not picatanny rails to swap and add optics. However, this is not a tactical gun and not needed. Later if money is no object, then add a picatanny and high end tactical rings if you must have them.
4. Used Nightforce 5-22 can be had for around $1400. You will be hard pressed to beat that scope, especially at your skill level now. Watch here, Benchrest central, 6mmBR.com and snipers hide for used ones.
5. I used to have my reloading setup at a friends house or in a cheap storage unit. Take the extra $ for reloading equip as you will need it. If not Black Hills offers the best and relatively cheapest ammo for you to try. Go to 6mmbr and look in the articles for plans on a cheap fold up reloading bench. I have one I use at the range and matches, works very well. It is built of the Black and Decker folding little work bench. Good solid press, set of dies, caliper, and other misc tools will be $400-500 starting so plan on that.
6. Several 100 yard ranges in PA, Williamsport, Read range and another out west. They shoot 1000 Yd BR and F class matches.
7. Some of the best LR smiths are in PA such as Kevin Cram (Montour Rifles here on this board), Mark King (King Custom Rifles) and Bruce Baer at Baer Custom Rifles. Joel Russo here makes an excellant wooden stock. you can google any of them.
For a great inexpensive stock look at the Stockys Stocks and his Bobby Hart LR stock. Costs $300, with chassis block already in it, laminated wood clear coated, sling studs and recoil pad. Drop your action in and go.