For target work with a 22 you need a scope with a
adjustable parallax. These come in two styles
adjustable objective (AO) or side focus. They do the same thing but side focus is a little harder to accomplish so it costs a bit more. At the short ranges that you deal with for rimfire shooting this becomes a very important feature if you're serious about bullseying the targets. What is does is let you correct out a distortion that might have the cross hairs appear just slightly off target. It's far less important for plinking or hunting.
The standard advice for rifle scopes is to budget as much for the scope as you do for the rifle. For hunting or plinking a 2-7x32 or 3-9x40 scope should be plenty. If you get into competitive target shooting the sky is the limit. I have a 6-20x50 on my target sporter and a 8-25x50 on order for my race rifle.
Bushnell can be a good choice but pick a good one. At least the Banner series. BSA has a reputation for falling apart really quickly and should be avoided. Some of the people on rimfire central swear by Mueller Optics' APV 4.5-14x40 but I haven't had good luck with my first Mueller scope. But they beat heck out of BSA and it might just be my eyes. Both the Banner and APV can be had for around $120 in silver or matte black.
Be prepared to spend another $50 on a better scope mount (not see through) and rings. You can't get a good cheek weld if you use see through rings. Oh and MidwayUSA has a guide for selecting the right ring height.
Then get signed up for some low cost Appleseed training.
Edit: My replacement Mueller Eradicator arrived today and meets my expectations.