There are a few options and I will suggest what I think are good calibers for starters and what they can be used for.
1. In some provinces a 223 centre fire is an accepted caliber for deer. And if it's going to be used for deer, it should be set-up for a minimum of a 60 grain bullets. And the Nosler partition would be the starting point. Then there is the federal 223 Power Shock 64 grain softpoint or the fusion 64 grain. Winchester makes a 64 grain power point. You could go up to a 70 grain bullet semi spitzer from Speer or with a 70 grain interlock. The last two would likely have to be hand loaded though.
2. Next would be a 243 using either a 95 or 100 grain hunting bullet. A good variety to choose from.
3. Stepping up a bit would be a 260 remington in a 120 - 140 grain bullet. Nice ballistics, but harder to find.
4. Last would be a 7mm08 in either a 140 or 150 grain bullet. With the right bullet this could be used for larger then whitetail deer.
5. Another round could be a 30-30 with 150 grain, 160 grain lever evolution or 170 grain.
With the exception of the 30-30 (mostly lever action) all of the calibers listed above are more readily available as bolt action rifles. Some are available in semi automatic.
Overall the best firearm to get a young person to shot well is a 22 LR.
