To move or copy a region, make sure you are in object mode. If you are using smart mode, the pointer must be in the lower half of the region to begin a move or copy operation.
Move the pointer into the region, use a Left-drag. The region will follow the pointer as you move it around. By default, the region can move freely along the timeline.
To move a region from one track to another, simply start a move as described above, but move the pointer into the desired track. The region will follow the pointer. Note that if you have other kinds of tracks visible, the region will remain where it is as the pointer moves across them, and will then jump to the new track. This serves as a visual reminder that you cannot drag an audio region into an automation track or a bus, for example.
To move multiple regions, select them before moving. Then Left-drag one of the selected regions. All the regions will move, keeping their positions relative to each other.
Sometimes, you want to move a region to another track, but keeping its position along the timeline exactly the same. To do this, use Middle-drag instead.