First question, how old and what size are the kids? 10,12,or 14 or small stature try .243. Older try 7mm-08, or .308 with the new reduced recoil loads. You then can use the reguler loads for yourself, and any of the kids that are of larger stature or when they reach maturity. At 15 my son was too small for anything bigger than a .243, then suddenly he had a growing sprut at 16 and immediately jumped to a 30-06. He is of small stature, and requires a youth stock ( he will probably never outgrow that stock). But he now has the weight, strength, and body bulk, to handle the heavier recoil of the 30-06 and his favorite, the 45-70 with 400gr loads (that thing with his hand loads kicks like my .338). Another advantage of the .308 is you can keep it for yourself after the kids have moved on, or give it to one of them later and it will make a good rifle for anything in the lower 48, except Buffalo. Last week watched a fellow take a Moose with a .308 using 180gr FailSafe bullet. Took three shots to finally drop the Bull Moose, but I blame the shooter not the gun.
As for accuracy out of the box. I have frequantely made 500 and 600 yard shots with a 30-06 Handi. I can hit 4" clay targets all day at 300 yards. I just put a scope on it and started shooting it. I did work up specific loads, but I did nothing such as lapping or excessive cleaning, or anything. As for cleaning I break down once everyother month or so and run a brush through the bore and then a few patches, then an oily one. That's all. But I never shoot it than twice with out letting the barrel cool. As the barrel heats up it will string them upward. But when hunting all shots are made from a cold barrel so thats how I practice. Two shots then put it down and shoot something else, then after it has cooled two more shots.