He's 20, and the scope/rifle package seemed appropriate for his budget. It came with a Weaver 3-9 "Caspa" scope. (?)

Its trigger is VERY nice but the stock seems flimsy. I already have him looking at nicer scopes (I've seen too many broken ones rattling uselessly on the rack at the gunsmith's shop) but other than sturdier glass and maybe a spare magazine "just because" what are some smart ways to make the decent rifle better?

Mechanically, what's the difference between the 110 and the Axis? What's most likely to fail due to cost-reducing shortcuts?

Also, what bullet weight does it eat best? He got a box of 150gr Hornady Whitetail ammo. I expect it to be fine, but figured I should ask.

Is there anything else we should know about this rifle?

Thanks!

I have a couple of those and I put a  Boyd's stock on one.  I did it more for comfort than any kind of accuracy implications but it was well worth the $99.  

Mine shoot better than me.

I told him about Boyd's. I really think the right laminated wood will look nice with the Cerakote he's planning. Back in the 90's Boyd's had a reputation for requiring a LOT of work for their "drop-in" stocks to fit and them not being very nice about complaints regarding the issue. Is that still the case?

Replacement stock as already mentioned would be #1 for me. The "10" series, has Accustocks and I don't think they fit the Axis rifles. If I'm wrong on that, find a take off and go that route for your cheapest and easiest improvement.
I'd shoot it with the scope as is until he feels like it's the limiting factor, no harm to be done using it unless it won't stay zeroed or something else mechanically is wrong with it. As they say, better to spend money on practice than upgrades until the equipment is actually holding you back.

I doubt there's any generic standard load that works well in them. Bullet weight really depends on his plans for it. I shoot mostly 147 "mil-spec" plinking bullets and then 175gr match bullets. If he plans to hunt with it then he'll have totally different priorities than my economical steel plate range fun at 100-200 yards with 147's and more precision oriented shooting with the 175's. These days, even 175 SMKs are considered a second tier choice. They're just cheaper than the real premium stuff and I'm not reaching out far enough to care about that last little improvement.

Surplus (or equivalent) 147 and 150 grain stuff is relatively cheap and effective. Most of it is reasonably accurate, unlike bargain 223 where a lot of it is minute of barn door. I reload the vast majority of my 308 ammo, but have been eyeing the slowly dropping prices.

Nope.  I have a Boyd's stock on a Rem 783 and one on an Axis and they both dropped right in, tightened the screws and done.  Mags fit and work fine.  And I can slide a dollar bill under the barrel.

My Boyds for Savage rifles were drop in. Heavy though, and probably not necessary.