Adjusting the beginning of a track on garage band
You need to grab the middle of the track, not the edge, and pull it back to the point where you want it to start. You can move all your tracks at the same time by selecting all of them with Cmd-A and then grabbing some place in the middle of the track and pulling them back.

If you're grabbing your tracks in the correct spot to move them on the track arrangement grid, your cursor will stay as an arrow instead of turning in to a bracket-type icon which indicates you're extending or shortening the length of the clips, not moving them on the grid.