On Shimano disk brakes, are both pads supposed to move?
Many mechanical cable-actuated disc brake calipers do indeed move only a single pad when the cable is pulled. I believe Shimano mechanical disc brakes use this design. Unlike rim brakes the disc pads can be adjusted to run very close to the rotor, so the active pad does not have to move very far to pinch the rotor against the static pad.

Hydraulic brakes typically move both pads. If you have hydraulic brakes one piston may be stuck. If your bike is new the shop you bought it from should sort it out for free or as part of a post-purchase tune-up. As you mention a pad adjustment screw I'm fairly certain you have mechanical brakes though.