I'll second the 25-06. It is very flat shooting, low recoil, and handles 120 grain bullets very well. Nearly anybody can learn to shoot one well with a little practice.

As far as .22LR, I think using one is illegal everywhere (or should be). A .22LR is good for close range varmints and plinking. It doesn't have the power or penetration needed to hunt large game with.

Texas allows the use of any centerfire rifle cartridge. I know people that use .222's and .223's successfully. But these cartridges have limited range for deer-sized game and limited bullet selection in factory ammo.