Henry, I will go along with the rest of the comments, all being really valid and sound.
I have a Howa 243, and it's a pussycat to shoot. And, it's compact and light enough to be a spectacular antelope and deer rifle.
As to the 308, it will hold its' own just fine down to 700yards, and as to the recoil, might I make a suggestion, from my recent experience with my son:
Last year, my son wanted to buy his 'own' first rifle. I am very sold on 30-06, and he agreed this was his first choice, with the anemic 300WinMag as his second choice.....

Visions of blurred vision danced in my head, and brought back memories of my Weatherby 300 magnum, which was a spectacular rifle, but I traded it out because of the recoil. It was just a stout, sharp hitting rifle, and not fun to shoot.

So last year was the great gun shortage, and what my son settled on was a Remington ADL in 300Winmag, with a synthetic stock. We looked for a 30-06, but they were nowhere to be found.
I warned him "This thing might be a real handful to shoot."
Our first trip out, we paired it with my Ruger 77, 7mm Mag. and my model 700, 30-06. An older version of basically the same rifle as my son, minus a half a soda can of powder in each shell.
And with the first BAM! from the rifle, he looked over at me, and grinned. A HUGE grin.

"Your turn, old fart....."
BAM. (Note; not BAM! but rather- BAM. ) The stock has some recoil absorbing features, and a decent factory pad. And I swear, the felt recoil was close to equal to my 30-06, and LESS than my 7mm mag by a good margin. The sharpness of the hit was just so much smoother than I could have ever imagined. Yes, it hits with authority. But not so much as to make you want to leave it in the case.
So, my point is; you may find a recoil deadening stock for the 308 which would make the rifle a very friendly partner to spend time on the bench with. The model 700 ADL is a standard rifle; no bull barrel, not real heavy at all. About 7.5lb for a 300WinMag. Price?? Even here in Peoples Republic of Kalistania, the cost was about $500, scope included. A decent 3x9 scope, nothing special, but was good to 250yds. has been replaced to compliment the rifles' ability to hunt well out to 500 yards or so.