I will agree even though I HATE 3-9x40mm scopes. But yes a 3-9x40mm is a good place to start. Unless you like me.
Below are my 2 Remington 22 lr rifles. The top one is a 513T with a Weaver Kt-15 15x40mm the scope cost me $250 the gun cost me $100. Bottom Gun is a Remington 511x with a BSA 4-16x40mm this scope set me back $75 the gun is worth aprox $250 to $300. They both will shoot extremely well. The 513 is sighted in for 100 yards and shoots 1" groups. The 511 is sighted in for 50 yards and will shoot 1/2" groups.
For low end scope tasco and Simmons are both of ok quality. Also look for used scopes on gunbroker and ebay. My weaver I got on Ebay.