I'm guessing that you want to use cheap milsurp ammo. Just be aware that the TRUE reason for the poor accuracy of milsurp rifles isn't the rifle itself but the inconsistent ammo.
How will you teach your son to shoot tight accurate groups if the ammo itself will only provide at best 8 inch groups at 100?
To save money on ammo AND provide the sort of accuracy that most of us desire I'd suggest that you get into reloading. I can reload .30-30 ammo for about 30 to 35 cents a round even up here in Canada where everything gun related costs more than in the US. And those rounds are as consistent or more consistent than factory ammo. This provides me with consistent shooting ammo that will truly deliver group sizes that are limited by my shooting rather than the ammo's performance.
A compact and simple reloading setup can be put together for around $200 and it'll all fit into a medium size storage tote when not being used. So all in all it'll cost you more to get started in reloading. And even with reloading you may find that milsurp is still a little cheaper. But the improvement in accuracy given by reloads will more than justify the trouble and expense.
Otherwise once the results are down to about 8'ish inch groups you won't know if you and your son are accurate shooters or that your performances are being masked by the poor ammo.
Here's an example of milsurp ammo performance. A year back I got a couple of Mosin Nagant rifles and a SPAM can of surplus ammo. The best I could manage at 100 yards were 8 to 10 inch groups with occasinal flyers out wider than that.
A couple of weeks back I finally got around to trying some Privi Partizan ammo and the groups immediately shrunk to 4'ish inch. Which for me and my old guy eyes using combat iron sights is about as good as I can get. Obviously if I'd had a scoped rifle this would have been even more of a dramatic change. Because of changes in this stuff are not linear even with a scope I'd be likely to only see 6 inch groups with the milsurp where we know that with good ammo something more down around 1 to 1.5 inch for 100 yards is standard.
So my advice is to start by figuring out an alternative ammo to the milsurp junk if you want to see your son truly perform as best he's able. And even for plinking if you want to cut ammo costs I'd say ditch the milsurp option and go with a nice .22 bolt gun. After all, rabbit stew is still game dining....