I say "get the .308 Win." if it is lighter than the .30/06 version.
These two cartridges are ballistically very similar out in the field:
The .308 is used in shorter action rifles that are lighter and handier.
It's all the power you "need" for deer hunting.
Surplus practice ammo used to be available inexpensively.
A 20 shot box of ammo in your pack will be lighter than the alternative.
This caliber is considered very accurate.
The .30/06 is ballistically like .308's big brother-- just about the same bullets, but sent a little faster.
It seems to shoot all the light and medium weight bullets a couple hundred feet faster than its little brother.
Heavy weight bullets are where it definately has a pronounced edge over .308
This is a much older, traditional caliber than .308 Win, loads of people shoot it in surplus rifles.
If you reload, you can download .30/06 to match .308, but you canna safely +p .308W to become a .30/06.
Frankly, my casual .30/06 hunting ammo is reloaded closer to .308W ballistics. This is great performance for iron sighted surplus rifles which will be shot at animals within a self imposed 200 yards. Further, my walking around blasting ammo has been influenced somewhat by Townsend Whelen's .30/06 small game load theory, which consists of a jacketed medium weight bullet going 1,500 fps. He reported excellent success with this downloaded ammo, and I'm experiencing the same for myself.
After you choose the caliber and rifle, set yourself up with a basic and inexpensive reloading outfit, and start to custom tailor your ammo to fit
your needs. You can use the $20 Lee Loader and experience all the thrills attendant with hammering on primers in your garage, and then moving up to an inexpensive O frame press, w/ RCBS dies and possibly the LEE scoop-type powder measures.
I suppose you realize that the possession of a .22 LR rifle is practically mandatory, for your marksmanship training? If you don't already own one, try to borrow one and then shoot it extensively at 25 or 50 yards, on paper. Target practice is
highly useful for hunters. For example, I have found Appleseed marksmanship training has been
extremely useful to me in the hunting fields.
More than you wanted to know?!

Good luck with this pursuit.