I leave the Chiefs alone as they should have good enough skills already. Concentrate on the petty officers and the Seamen, Torpedo rooms, Watch, Sensors and Command should get priority in any boat.
Get the PO's up to 1st class and watch their skills improve along the way, the seamen can be promoted to petty officers and their skills will improve when they are promoted enough.
If you need better gunners promote the best two in the damage compartment, promote them high enough and you can select guns as their skill, mechanical, watch and guns are the priority for that class.
Look for the little black numbers, it will say something like 0 of 150, when it reaches 150 of 150 promotion is available for that crew member.