Tsavox - For your captain's skills, I believe they just unlock as you level up based on the skills you are taking and your career. Perhaps someone else can flesh out that answer.
Each of your bridge officers has one ground skill and one space skill _per rank_ (Ensign, lieutenant, etc). In order to be able to use their skills, they have to be promoted to the appropriate rank. For the space skills, they also have to be assigned to a station of the appropriate level on your ship. So for instance, a tactical lieutenant will have access to two skills (ensign and lieutenant) if assigned to a lieutenant station. If she's on an ensign station, only her ensign-level skill will be available.
You can replace your bridge officers' skills by buying them from the skills trainer.
Happy to be corrected if I got any of this wrong. =)