308 rifles are far more common
I don't see that at all. Maybe it's a regional thing but 30-06 outnumbers 308 at least 10 to 1, or more, around here.
Go to a good gun store and shoulder a lot of rifles. Write down the ones that feel good and the ones that don't feel quite so good. I personally prefer shorter rifles, carbine length, because they "feel" good to me but I often hunt with longer guns as well. I am 6'1" but really like the way a carbine shoulders.
As far as caliber goes there are many that will do the job on deer and hogs and not hurt your shoulder. If I had to have just one hunting rifle and I was in any way worried about my shoulder then I would stay away from 30-06 or 308. Not that they are brutal but there are some fine calibers that have lower recoil and will do the trick. You CAN buy reduced recoil rounds in 30-06 (maybe 308 as well but I don't know). I have never personally fired any but my hunting buddy got his grandson an '06 and bought the reduced recoil loads. I hunt with a 30-06 quite a bit and have killed many deer with several different guns in that caliber. It is probably the most ubiquitous caliber in the USA for "big game" hunting.
The 243 is a great option with reduced recoil. Flat shooting at most hunting distances and plenty capable of killing deer and hogs. Ammo is everywhere.
Lately I have been hunting with a sporterized Swedish Mauser in 6.5x55 and I cannot say enough good things about the gun. Short, light and extremely accurate. I have dropped every deer I have shot with it and a 14 year old girl got her first deer this year using the gun. Very light recoil. Ammo is not available at Walmart but can easily be had online or at most of the Bass Pro/Cabela's type stores.
The 25-06 is a good caliber for deer but I have never shot a hog with one. Light recoil but ammo isn't nearly as common as 30-06 or 243.
I don't personally own a 270 but I have fired a few of them. The recoil, to me, is close to a 30-06 but that could just be the guns I have fired. Plenty of gun for any animal you will likely shoot. Ammo is plentiful.
The 30-30 is a tried and true deer gun that has very light recoil. It may be more of a challenge if you will be shooting at distances over 200 yards as the bullet drops off quickly at those distances but it is a great gun at 150 yards and in which is where the vast majority of deer and hogs are taken. Short, light and plentiful ammo.
You can certainly kill deer and hogs using the 7.62 or the .223. Many will say not to use the 223 (it is illegal in some areas) but it is a decent deer cartridge and you know how light the recoil can be. Hunting ammo is available though not all that common in normal gun stores.
As far as a brand I really don't think anyone can help much with that. My favorite 30-06 is a plastic Savage with the accu-trigger and the Swede has a custom wood stock. I have a BAR and several other 30-06 rifles that I rarely take to the woods. Tikka, Browning, Savage, Ruger, and plenty of others make good quality guns that will come in at around $400-$600. Spend another $400 or so on a quality scope and you will be set to hunt for the rest of your life with ONE gun. Yeah, Right!!!!