OK, so we've got:
1. New
2. Under $200
3. *Mostly* target shooting, but some hunting, and most if not all target shooting will be 50 yards or less, sounds like...
First off, in magnification, for those purposes, I'd go with a fixed 6 power, or one of these variables: 2-7, 3-9, or 4-12.
One scope in that price range is the Bushnell 'Banner' 4-12x40 with adjustable objective. These run $85 - $110 plus shipping.
For a little more however (right at $150), the Vortex 'Diamondback' 4-12x40 is supposed to be a pretty good scope.
You might also look at the Mueller 3.5-10x44 Sport Dot. Or a Swift Premier 4-12x40AO
I like 4-12 best for that use (mostly target shooting) - the 12x will really help at 50 and 75 yards, but you can still dial down to 4 for hunting (4x is about the 'maximum minimum' I'd want to have on a .22 hunting scope).
Get GOOD, low steel rings.
Oh that FX-1 Leupold 2-7x28mm sounds good at a little over $200. Also, for about $100, the Weaver K-series rimfire in the same config is pretty good (2-7x28mm).
To sum up:
--If it's mostly for hunting, then I'd get a 1-4, 1.5-4, 1.5-5, fixed 4, 1.75-6, or 2-7
--If it's mostly for target shooting, then I'd get a 2-7, 3-9, fixed 6, 4-12, 4.5-14, or even 6-18.