The process can be different depending on where you are; the process weblance described above only works if your local law enforcement will sign off. If they won't (like in my neck of the woods), then you need to purchase the suppressor through a revocable trust. In many ways the trust is superior; it doesn't require photos or fingerprints and it allows more than one person to be on it, meaning more than one person can possess the NFA item. Also, you can designate a person to receive your suppressors upon the event of your death. On my NFA trust I am the settlor, I am the first trustee, my wife is the second trustee, and my father-in-law will receive my suppressors in the event that both my wife and I die. The trust can be amended at any time, but I recommend you get advice from a lawyer or other qualified person to do that.
The main problem is getting a trust. Up here (where almost nobody can get a sign-off from local law enforcement) most dealers make you get your own trust, which can run in the hundreds of dollars. I'm spoiled because my LGS offers a trust for free and it's super easy to set up.
As for the suppressor itself, that's a different issue. I'm a fan of AAC and I think the most versatile rifle suppressor on the market is the SDN-6. It's a quick-detach, full-auto-rated .308 suppressor that is optimized for subsonic 300 Blackout use (or any other 30 caliber subsonic round). Most people (including myself) won't ever shoot it on a full-auto, but that tells you how strong it is; it will easily handle any cartridge from .30-06 on down. I got one for my 5.56 AR because it's almost as quiet as a 5.56 suppressor but it's so versatile.
If you only plan to use it on that one rifle, I'd go with the AAC Hunter; it's the same as the SDN-6 but it has 5/8x24 threads (same as your rifle) and it will be lighter, cheaper, and slightly more accurate because it's direct-thread instead of quick-detach.
If you never plan to use subsonic ammo, go with the AAC Cyclone; it's a larger thread-on .308 suppressor that will be slightly quieter than other two with supersonic loads, but louder with subsonic ones.
0331: "Accuracy by volume."