Federal Premium 180 gr in either Trophy bonded tip or Nosler partition. A moose will not know the difference.
Same bullets in the 160-165 range for deer
You will get lots of different opinions on the exact ammo to use, and you probably can't go wrong with any of the choices. However, you need to try them in your gun and see which one shoots the best. The choices are generally not inexpensive, but you can use cheap stuff for plinking, but take the "go to" ammo when on a hunt. It is cheap insurance.
Yes, you can hunt with Remington "core-lok" ammo and put in the boiler room, they animals will all be dead. I use it on Texas whitetails. But on bigger deer and elk, I like a tougher bullet in case the shot is not ideal.