On Shimano disk brakes, are both pads supposed to move? I have Shimano disk brakes (cable operated, no hydraulics), they are still pretty new.

I noticed that under braking, only one of the brake pads moves inward.

Is this how they are supposed to work? Or are both pads expected to move?

I do know there is an allan-key screw that I can move to adjust the brakes, but I don't know if both pads are supposed to move when braking.
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.