As you mentioned that you did NOT intend on spending big money, I would suggest to you the Ruger 10/22. There is an endless supply of upgrades available as your budget allows. Two quick ones that you may consider:
These sights allow simple adjustment for windage and elevation. Tech-Sights give you a very similar sight picture to that of the Springield M1A and the AR-15 rifles.
One of many items available at
1022central for $149.00
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I am not associated with any of these products or companies, just making a suggestion. I have a couple of 22s that I enjoy shooting with my son. Nothing fancy, keeping the cost down, guns & ammo, allow us to shoot more often. You should be able to find a new Ruger for right around $225.00, which will shoot just about anything. I did a quick
search and found the lowest price on a Anschutz to be $755.00 with extra magazines to be equally higher priced.
You can read find a lot of info on inexpensive 10/22
here. Just enter LTR in the search. May not be what you are looking for, but maybe in the budget for others reading this.