While overall I agree with everyone here about the bolt gun for hunting here’s some advice ion the AR end of things:
– don’t build it completely from scratch. Buy a lower, buy a lower parts kit, and then buy a complete upper. Modify from there as you see fit, but by doing this you have made 3 purchases and have a complete rifle.
– 5.56 is the best caliber for your application. It is abundant and cheap Although currently this panic buying has screwed that up somewhat it will return to normal eventually. Whoever tried to up-sell you on 6.5 is full of it. 5.56 is effective out to 600 depending on barrel and ammo. 200yds is easy mode for 5.56 even using cheap ammo.
– Contrary to one comment above, a $700 AR is plenty accurate and will easily exceed your capability as a shooter. So will a $500 AR. I would not recommend that price point for bench rest competition shooting. That is not what you’re doing with it as you indicated.
– For a quick and easy intro to the AR world, Go for something from Palmetto state armory. Be advised their “Freedom” brand is where the quality starts to decline sharply. Use their PSA brand and you have a very respectable entry level AR.
Best of luck hope some of it helps