Wire gauge for subpanel
Here is a link to south wire voltage drop calculator your voltage drop should be less than 3%. I entered 240v 160' and 80 amp load since you did not give a value. The calculator provided #3 copper. You would want to use UF cable for direct burial. If you are putting it in conduit you would want to use THWN. Schedule 80 PVC conduit would need to be 1-1/4" for 3 wire or 1-1/2 for 4 wire.
In your sub Panel the ground and neutral needs to be isolated many times they are tied together at the factory with a large copper wire connecting the 2 grounded / grounding buss bars. Other than that you should bury this a minimum of 18".