What do you have and use yourself? For hunting? For target shooting? What is available for him to try before you buy anything?
A deer/hog gun is not often used much. Not very many people get the chance to shoot a lot of game. In my state, I have to win a lottery just to get a tag for at most one deer. Some deer rifles go a lifetime on one box of ammo. On the other hand, shooting no more than a handful of cartridges annually is not the best way to develop skills. What's more, shooting a costly big game cartridge for practice is expensive. Even so, the cost of a second rifle would also pay for a lot of practice ammo.
I don't recommend cheap guns. The problem with spending a lot on a good gun and optic is that you want to know what you want before you do that and if you're starting out, you just don't know. What you do know is that you don't want the cheap stuff. Buy better stuff and spend more or less based on how certain you are that it is what you want.
For smaller kids, the low cost and low recoil of rimfire cartridges are compelling, but getting the rifle to fit is still important and often overlooked. Your son is already adult-size and is apt to fit off-the-shelf rifles as well as anything not custom fit for him. The cartridges best suited to training are 5.56x45 and 7.62x39. Right now, the next most affordable cartridges are twice as costly. In better times, there are more choices that are affordable to shoot. The 5.56x45 is always going to have an advantage of lower component costs because the smaller bullets use less material, and the cartridges use less powder. 5.56 also recoils less which makes it a better intermediate step between a rimfire and a heavier deer cartridge. The 7.62x39 could be said to be a deer cartridge (with a suitable bullet design).
Because a training rifle in addition to a deer rifle adds substantial cost, there are some things to consider as alternatives. First, if you handload, you can use a big-game cartridge to introduce an inexperienced shooter by using reduced loads -- particularly loads made with Trailboss powder. With Trailboss, you can make .30-06, 458 Win Mag, and 375 H&H loads that are comfortable for any child to shoot provided the rifle fits them within reason. Of course, this makes the most sense if you already have such a rifle and the shooter being introduced is an inexperienced older teen or adult that will with some practice quickly graduate to being able to use the rifle with loads powerful enough to hunt with. Even if you don't have such an overpowered rifle, reduced loads can help someone get up to speed before they shoot full-power .308 or 6.5 Creedmoor for example. What this won't do is provide an abundance of low-cost cartridges.
Another alternative would be to consider a rifle that can be chambered for multiple cartridges. With an AR, you can shoot 5.56x45 for training and practice and with a simple change of the upper and sometimes the magazine, configure it for another cartridge that is ideal for deer or hogs -- 300 BLK, 300 HMR, 6.5 Grendel, 450 Bushmaster etc. etc.
Basic rifle skills are more important than killing one or two deer. First, they will be indispensable for any amount of hunting, a few deer or many over one's lifetime, and second, they may be useful for many other reasons. That's why I would consider a training rifle even if it were in addition to the hunting rifle. While an additional rifle may substantially add to the overall cost, it is an asset that retains value. The excess cost of big-game cartridges spent on paper targets is a total loss.
