I am going to offer my opinion on the three cartridges you mentioned. If you plan on hunting deer/antelope/sheep at longer ranges go for the 7mm Remington. If you plan on doing that plus elk and moose, consider the .300 Winchester magnum with heavier bullets. If you plan on hunting big bears or African game, go for the .338 Winchester magnum.
Really, any of them will do the job, and the .300 will handle everything well, but it comes down to what you feel most comfortable with. Typically, your looking 140-160 grain bullets for the 7mm, 165-180 for the .300 and 225-250 for the .338. Velocities will range from around 2700-2800 up to 3200-3300 fps, depending on your cartridge. And your range will allow pretty much dead on holds out to 300 yards with all three. In my experiance, the .300 kicked the worst. It was a much quicker, snappier boot than the .338's. They are a slower, heavier but easier to handle push. The 7mm was similar to the .300 but with less force.
Ammo availability for the 7mm and .300 are greater than the .338, but I think they give false impressions. I don't think the .300 should be shooting a 150 grain bullet. And I don't think the .338 should be shooting a 180 grain bullet. These lighter bullets will give you a flatter shot, but they don't do the cartridge justice. Find a rifle you like, see if it's offered in your cartridge and go from there, either way your picking a winner.