I use A to select all my model parts, but when I hit G to move it only the grid moves. How do I correct this?
Are you currently looking through your camera? If so, by selecting everything in the scene you also select the camera. By then using GKey to move, you're moving all your parts and the camera as well, so it'll appear that only the scene grid is moving.
Try locking the camera in place (the X, Y, and Z locations on the N Panel) and see if that helps.
You have the view locked to an object. Press N to show the 3dview properties region and find the View panel, in there you will find the Lock to Object option.
Click the little
X to the right to disable the lock view to object.