Everyone remembers their first hunting rifle.
Throughout my hunting career, I've shot all three of these rifle calibers. If you ask me, they're all really similar in a lot of ways. However, let's take a look at pros and cons of all three hunting rifles.
First and foremost, I own two of these guns, and the .223 is one of them. I take my single-shot, low-recoil, .223-caliber, breakaway rifle hunting sometimes and have been very successful. The low recoil makes it great for a new or young hunter who worries about bruising their shoulder or securing the gun. Most .223 rifles are very lightweight, so just about any hunter is able to carry them.
Unfortunately, one con of this caliber is that the hunter needs to make a great shot. That said, I haven't had any deer run very far, but I also only took shots under 100 yards. There just isn't a lot of knockdown power with this caliber.
It's a great small-game hunting rifle, but you're certainly limited. In fact, some states don't allow hunting big game with a caliber this small, so be sure to check local hunting regulations.
I own a Savage bolt-action rifle chambered in .243. I kill deer with it. Simply put, if I picked one gun out of the three listed to hunt deer, this is the caliber I would choose. For a younger hunter this is the perfect combination of knockdown power and low recoil.
Another great option is the 95-grain Hornady cartridge called the SST Superformance. Hornady boasts a 2908 fps at 100 yards and 2648 at 200 yards. Both the Hornady and the Remington are great rounds, sufficient at long range for any young hunter. They make some reliable .270 and .30-06 rounds as well, but that's an article for another day.
The 7mm-08 is a great deer rifle, but I think it might be too much for a lot of young hunters. You can make adjustments to lessen the recoil, but a .243 is probably a safer bet. What's interesting is the 7mm-08 is essentially a .308 converted to hold 7mm bullets. It's doesn't kick like a 7mm Remington Magnum, but it certainly has more bite than a .223 or a .243.
What's most important is taking your young hunter target shooting first. Take them to the local gun range and let them try out each caliber. Then take them deer hunting with the one they feel most comfortable hunting with. Ask other hunters with experience, too, as advice from other hunters is always valuable.