Just saw this in a local hobby shop. The box says it's motorized. If I'm not mistaken, its a Tamiya German Panther 1/35 scale. Does it move in 4 directions? Does the turret spin around? Does it have a remote control? How many and what kind of batteries does it need?

Depends on whether or not it is a newer motorized vehicle or not. Tamiya produced several motorization types.
Early was just a gear box and the tank went forward only.
A wired remote was included with a dual motor gearbox so you had left/right/forward/back.
Their initial M1 came with a 3-motor set up that also allowed the turret to traverse as well.

Lately, they have produced the true RC 1/35 scale kits. I've got the M1A1 with wireless RC. It can go forward/back/left/right and includes the remote control. Usually uses D cell for the tank, and AA batteries for the remote.