With $2g you can easily do an AR10 / AR308. You can get mil suplus ammo to plink or nice match grade to reach out and touch someone at 1000+ yards. Of course you'll ned some good glass to that.
MJ's suggestions are great too, if you start out with a complete 5.56 you can opt for a 300 blackout or 6.8spc upper later.
So you can a decent .308 and some ok glass or a very nice 5.56 that decked out / fully loaded for $2g. My last AR15 was 1300 + 550 red dot + 45 sling + 50 trigger upgrade + 30 bad lever comes up to just shy of your budget. The 308 platform is even more expensive, so you'll have to scrimp a little more to make your budget and high quality glass an be just as if not more than your actual gun.
Assumption is the mother of all ****ups.
Know what you know, Know what you don't know. -Paul Pedzolt
"Jack of all trades, master of none, though oftentimes better than master of one" -Mynshul