It is clear from your post that you don't have much, if any, experience with centerfire rifle shooting. Recoil can be measured by mechanical means, and using those means it will be clear that the .308 recoils substantially more than the .223 -- but that information really doesn't tell you much. As RustyShackle has said, the effect of recoil on the shooter is wholly subjective. I can well recall when as a kid I first fired a .30-30 rifle. The recoil seemed unbearable to me at the time; now, it seems barely worth notice. Experience has taught me not only that the .30-30's recoil is mild compared to that of many, many other big game cartridges (including the .308), it has taught me how to handle the recoil of even much heavier rounds without discomfort.

If I were teaching you, I would start you off with the lowly .22 Long Rifle. Not only is the .22 one of the most accurate and useful of all cartridges, it is also quiet and just about totally recoil-free, and allows an inexperienced shooter to work on and master the basics of rifle-shooting accuracy without the distraction of recoil. Then, ideally I would move you up gradually -- maybe to the .223, then to the .243 -- and let you accustom yourself to some recoil before getting in to kickers of the .308/.270/.30-06 class.

In keeping with that philosophy, the .223 might be the right choice for you to make; not only will the recoil be much milder, but the ammo cost will be substantially less than for the .308. These two factors might allow you enough comfortable, affordable shooting to turn you in to a real rifle-shooter, whereas jumping right in to the .308 could make you both too gun-shy and too broke to go very far. Later on, if big-game hunting becomes a real possibility, a step up to something in the .308 class would be appropriate, and the transition from one level to the next much more manageable. I am thinking of a couple of people I know who decided upon a .44 Magnum as their first handgun, and who soon swore off handgun shooting altogether and lost substantial money selling their expensive revolvers. That's NOT a result you want!