Add 1/2 inch plywood if your joists are 16 inch on centers AND you are going to use 1/4 inch concrete backer board (Hardibacker)
Use 5/8 inch plywood if you are on 24 inch centers AND you are going to use 1/4 inch concrete board.
Use 3/4 inch with no backerboard on 16 inch centers.
3/8 inch plywood is not enough even if you use backerboard. Do Not consider it.
Lay the same direction, but stagger the joints.
Use the screws at the joists, but use 1 5/8 inch screws at the joints.
You could use some construction adhesive (like PL400) between the sheets, but do not glue the sheets together with wood glue.
Good Luck!
Dodgeman