Naval skill is how the ship is handled. It helps escorts find subs, it helps ships avoid hits, it affects most of the combat related things that happen, increasing a chance to cross the T or attain surprise.
Leadership adds experience to the crew with various actions. The higher the leadership, the more quickly the crew gets better experience. Think of it like drilling more effectively.
Aggression will decide how long the ship stays in the combat and how it reacts to other ships/TFs, but that is mostly a TF leader skill.
Air helps air ops coordinate and fight effectively, or search, or recon.
Admin helps the process of loading/unloading ships more quickly.
Inspiration does nothing for ships or TFs. (It helps raise morale and AV adjustment with land units and air groups, and helps get more planes in the air).
For carrier captains, the primary/secondary skill is Naval/Leadership? Naval helps it evade battle in case of surface engagement and Leadership for quicker experience gain. As far as carrier TF, Only the TF leader needs a good air skill?
for the tf commander, you would want high inspiration as more planes would fly.
Life is tough. The sooner you realize that, the easier it will be.
Currently chasing three kids around the Midwest.