I agree with the Savage choice.
They'll be a "starter" rifle in your price range.
I think caliber is pretty irrelevant at this point. I would go with .223 since you do not reload, and ammo is cheapest (or will be, when all the crapola dies down).
I wouldn't bother with a "better rifle and lesser scope", because there really isn't anything "less" in the $500 range for both.
The advantage with going Savage is that when you're ready to move up, you can upgrade to an aftermarket barrel, stock, and trigger- PROVIDED you go with a Savage 10/11 or Stevens 200 (same action). If you go with a "package" Axis model, you won't be able to do that.
These can be had for about $330 online. You'll need an inexpensive optic for about $150, then... If you can, try to spend a bit more, in the $200-$300 range for the optic. The .223 is low recoil, so a cheaper optic should hold up.