The wood one is a Remimgton youth model (several years old now).. The other is a Ruger. Both are very afordable. Both are .270 caliber. I use the 130 grain slug for deer. Those elk were taken with 150 grain slugs.

I own a .243Win, .25-06, and a .280AI. None of them really have that much recoil in my opinion. The .280Rem will have the most when using heavy bullets but it is still not bad.
I have killed large western Mule deer and an Elk with both the .243Win and .25-06 and none of them required second shots. The 7mm-08 is another great cartridge that would do well on deer or larger game.
As far as the Rifle's are concerned Tikka makes a nice rifle but the only bad thing about them in a .243 is they use a 1:10" twist so the 100gr bullets might not work as well. All the other calibers the twist rates are about right for caliber. Ruger makes a nice rifle as well as others. The TC Venture just had a recall notice so that might be a concern for me. CZ makes a very nice rifle as well. Try and hold as many rifles as you can and make sure they fit you both well before you purchase one. A properly fitted rifle will certainly help with recoil. And if you do find recoil to be a problem then you can always install a limbsaver recoil reducer buttpad that will help reduce any felt recoil.

I'd opt for either .243 or 7mm-08. Both great guns, low recoil. Should fit the bill nicely. If you go up, look at the .280 rem.

If you ever plan on going after any bigger game id seriously look at a .280 works great on elk. For your wife a .243 is plenty plus you learn to take better shots with the smaller calibers. If u never plan on going after bigger game a 25-06 is a great caliber also ammo can be a bit harder to find and more expensive around here at least.