You're getting great advice here, drtony. I'll corroborate the idea that a stout 180gr or a very stout 165gr bullet from a 300Wby is going to be good medicine for elk. I'll have my 300 with 180gr E-Tip handloads should I get the chance to go after elk in the future. It's on my list, but other things seem to keep bumping it down - college costs for my kids, orthodontics for the kids, new truck for my wife (I see a theme here...).
As for recoil, I'll offer a couple of thoughts. First, noise is a huge factor in recoil. I've found a lot of folks get noise-based flinches, and they carry over to such hard kickers as 22lr and 243Win. If you're not wearing double protection, I'd suggest you get some good plugs (30-31db reduction) and some good traditional muffs (>30db reduction), rather than electronics. Wear the plugs and muffs together, and get situated behind the rifle in a comfortable, not too bent over position. If you need to adjust the seat height or the rifle height, use a cushion (I suggest a padded boat cushion) to raise your seat (or more than one if needed) and raise the rifle with a rest or sandbags. Getting in the right position does wonders for mitigating recoil. The range I frequent has too short a bench for the seat height for me (I'm 6'2" and a little long in the torso). I have to raise my rifle a bit from the surface of the bench top to get really comfortable.
The last tip is to get a recoil pad for your shoulder. PAST makes some excellent ones. I use an old bath towel most of the time. I figure it simulates the thickness of my hunting clothing during the colder part of the season (when most of our deer activity occurs here in northern Alabama), so it works well for me. It also pads my shoulder from the barrage of rifles and ammo I take to the range when I finally get time to go shoot.
Beyond that, start with the 150s in your 300, as the recoil will be 5-10% lower than the 180s. Expect it to be a sharp jolt at first, but over time you'll get used to it. Just work on the fundamentals, and swap out to smaller stuff with lower recoil frequently so you can maintain your form and trigger pull and breathing.