Bunches of great advice already given.
My 2 cents - a bolt action 30-06 or 270. I'd get a wood stocked one if you really want to impress but I wouldn't think anyone would frown upon a synthetic stocked bolt gun. Oh, and you cannot go wrong with a Remington 700. You can find ammo for either of these cartriges in any store that sells ammo.
Around here you would definitely turn off the old timers if you showed up with any caliber AR to deer hunt.
Would be great to start with a .22 of course, but if you can't afford all that, you'll be alright. I was taught to shoot on a .270 and I turned out fine. That was when I was 10. Dad didn't have a .22 or that's what he probably would have taught me with.
Best option ,like has already been mentioned, would be to have fil go shopping with you. Even better would be to have him help you sight in a scope and such.
Guys bond buy doing these kinds of activities.
Bottom line- follow the advice in this thread. It has been a great read for me and there is a lot of great advice here for the new hunter. Only thing I very slightly disagree with is some of the caliber recommendations. And that's only because some of them would be hard to find in a pinch.