I'm rather new to sh3 so better listen to the veterans here but this is how I upgrade the men:
First I let the navigation officer become my repair expert (the LI could do this job as well but there has to be an officer on the rudders so I ignore him).
Then I get a new officer to make him my machinist.
Then I give the sonar knowledge to a NCO.
After that I make the weapons officer an torpedo expert to reduce the loading times. Most of the time I have him in the front restroom so he might also become a medic later to reduce refreshing time of the men.
The watch officer gets deck gun skills.
And then... well, I never lived long enough to decide who to upgrade after that.
