I have to agree with jimmythegeek on the Marlin 795. It is as accurate, many say more accurate, than the 10/22 out of the box and the 10/22 is out of the OP's price range even before you get it out of the box. You can put a scope on the 795 or a set of peep sights (ex: Tech Sights) for less than $200.
$125 (795) + 60 (Tech Sights) + 15 (sling) + 5 (elbow pads) + 20 (bulk box of ammo) + 20 (RWVA membership) + 80 (local Appleseed weekend) = $325.
If there is a more accurate setup and the training to make proper use of a semi-auto then I would sure like to know about it.
NRA Life Member
"There are some ideas so preposterous that only an intellectual will believe them." - Malcolm Muggeridge