You don't mention if your Weaver is a vintage USA model, or one of the newer made overseas models. Just curious.
Irregardless, there are bound to be LEFT/ RIGHT and UP/DOWN markings on the dials. Follow that and don't watch the reticle move in your sight picture. As a broad comment, most scopes adjust bullet strike, just like the dials move.....screw the elevation dial "in/down" and the bullet strike does same. Same with windage, screw it "in/left" and the bullet strike moves left.
Failure to zero can be many things...loose mount system, bad scope, loose action/stock screws. As I remember, Sav 340 used a funky sidemount like arrangement, be sure and check your mount screws.
Sounds like you can shoot, and the rifle will too. Good luck with it. I guarantee you will shoot the scope better when you get it all worked out.