I know you said rifle in your OP, but for small game and birds you want a shotgun, not a rifle. You will give up hunting very quickly if you decide to hunt bird and rabbits with a rifle.
I just bought a remington 870 12 gauge. It was $250 new at Cabellas.
The 870 has had essentially the same design since the 1960s. It is a very easy first gun to own. You will be very happy with it, especially at that price.
There are literally millions of 870s in circulation, and if you search for any gun reviews, the 870 consistently is rated as one of the best shotguns at a fraction of the cost.
As for size. At your age, you are ready for a man-gun, don't bother with the 410 or 20 gauge. the 12 gauge you will have for the rest of your life.