Can I run 6/3 cable from my main panel to a sub panel located in the same garage? And does th 6/3 wire need to be covered,it's going up the wall across the ceiling and back down the wall to the other side of the garage and from there it will enter the house near the sole plate.

The cable can be surface mounted using the proper straps or stapels every 4.5 feet as you explained. It would need added protection if it can be subjected to any physical damage, which I doubt, unless, for example, you hang rakes and shovels on the wall in the garage over the cable.

If the cable is exposed low on a wall, that area may need conduit to protect the cable.