If you can get any of the "match grade" variety of the heavier rounds, that should be more than enough to get you started. If you end up really liking it (long range shooting can be very addictive) it is worth it to look into hand loading down the road. How is access to reloading components in NY?

It's a little easier with .30 caliber rounds, but long range shooting can still be a lot of fun with the 5.56, and you can learn all the principles on a 5.56 that you can with the larger calibers. I generally keep my 5.56 inside of 600 yds (it can definitely be hard to find places to shoot that far), and take .308 out to 1000. It's certainly a fun challenge regardless of caliber. If you are shooting outside a designated long range shooting range, steel targets are very handy for gauging hits at longer distances; a spotting scope and fresh coat of paint on steel targets helps a lot to confirm hits.

I forgot to mention, Dishonor Code does a lot of long range shooting, much of that in 5.56. He's another good resource on the board for precision marksmanship at distance. I think you'll have a lot of fun with it though, I love estimating range of an unknown distance target, making the calculations, dialing in the dope, and sending a round. The challenge of getting first round hits, and second round corrections is a blast.

You'll have to keep us posted on how things turn out. Sounds like you know your stuff and have some good kit to work with!