I don't consider myself particularly 'recoil sensitive' and I shoot my 3 rifles fairly regularly. They are a .243, a .270 win, and a 30-06. My .243 is in a Win Model 70 with a Hogue overmolded stock that I added. It has virtually no felt recoil to me.
Based on what I have read here, the .243 is a great deer rifle and can be used on elk at reasonable ranges. I know for a fact that .243 win is almost always in stock at the local Walmart and everywhere else that sells much ammo.
So, my recommendation would be the .243 AFTER you have a good .22lr for lots and lots of practice