I have two 338s, a Savage 112 Magnum Target and a 110 Elite Precision. I shoot these at 1,000 to 1,800 yards, with plans to go farther as we are good to 2+ miles at the private range we shoot at.
I started out buying a lot of factory second 250s and deman 250 SMKs because they were inexpensive. I don't load them much now other than for low-cost fire forming, although they both shoot pretty well.
There are two loads I shoot a lot, and both are good loads. Please note my OALs and that I HBN coat my bullets so jumping in on these loads at shorter OALs may not work well for you. I use UG, Fed 215 and CCI LRM primers, UG preferred frankly; lower ES.
#1. Hornady 285 ELD over 94.0 N570 to an OAL of 3.809" in the 110. 1/3 MOA shooter, a bit flatter than the SMK and shoots well. 2,869FPS, 15 FPS ES. 1/3 MOA load any day you choose.
#2. 300 SMK over 97.5 RE33 (now gone invisible) to an OAL of 3.752" in the 110. 2,872FPS, 20 ES. This load is a bit much with Bergers 300s, I would back off 1.0 to 0.5 grains at least. Sometimes you can find 300 SMK demans at American Reloading for a deecnt price.
I am extremely happy with each load to 1,800 yards. For longer I will look at the 300 Berger or 300 A-Tip.
OK, cases. I have never used Lapua cases or Nosler cases but I like Peterson, PPU and S&B cases from their ammo. Looking at Grafs, the "Major US Mfg brass" is Peterson bulk packed. I *love* my PPU brass. When it was $60 a bag it was a gift. I get at least 8 full power loads per case (so far). I do clean up the flash holes and regularize the primer pockets but that's what I shoot 1/3 MOA groups with. Now Grafs and others have doubled the price, they aren't such a steal but don't turn your nose up if you see any at a good price.
That's what works for me. BTW, if you have a problem finding powder LRM is a good choice, slower than RE33 and needs tight neck tension (my default is 0.002") and magnum primers but can shoot very well. Somehwere in the region 103-105 grains worked for me in the 112 with 250s.