How to determine the supply side & load side of a cable/wire ?

It would be helpful if you could tell us what you are doing. Usually it is the black wire but could be red, blue, brown or any color not permitted as a current carrier, e.g. green. Using a VOM, set the VAC to 150 or more, remove the switch or outlet and connect one end of the test leads to black, the other to ground. If it reads voltage, that is the supply. However, if a switch loop is used, the white lead may be the supply. 3 or 4 way switches complicat things more. What is the situation you are trying to figure out?

You are looking at "wire size" incorrectly.

The size of the wire you need is determined by the amperage the wire can carry.

Different wire sizes are "Listed" for the max. amperage they can carry. For example, 14 AWG conductors can carry 15 amps. 12 AWG conductors can carry 20 amps and 10 AWG conductors can carry 30 amps.

Once you determine the amperage required for the load you want to supply, you select the proper breaker for the amperage that you need. For example you might need 20 amps, so you select a 30 amp breaker and a 12 AWG cable.

Is that what you are looking for?