Staff skills improve automatically depending on their talent and whether they are assigned to one of your facilities.
An assigned staff member with 8 talent will gain one star each season on the facility he is assigned to. His other skill remain unaffected.
A reserve staff member will gain skill 3 times as slow but on all fields. That means such a staff member with 8 talent will gain a star on all of his skills every 3 season. This will not necessarily happen at the same time.
There are no intermediate levels, so any staff member with a skill of 5 on a specific facility will have exactly the same efficiency. There is a hidden learning progress though that keeps track of how close a player is to advancing to the next level. If you assign or unassign staff members, their progress will be kept.
