Ground: Your main character's skills are dictated by their kit loadout. In that sense, you don't internalize your skills the way your BOs do. On the flip side, you can carry multiple kits and switch mid-mission to suit your situation accordingly. Oh yeah, and you do gain some skills based on your class.
You have to in order to rank up. It's on the same page as BO's.
Hit K. Select the Status tab. It will ask you to select an officer from the list on the left. Your CO is at the top and trains just like any of the other BO's which are listed below your ship name (which is below your CO name).
Be careful though cause your CO has a much longer list of skills. Make note of the slider on the right and make sure you review them all before you commit.
I know you can train you BO's and give them new skills but...can you do the same for your main character?
That doesn't sound stupid at all, but no you cannot train your toon as you can your BOs. You give your toon skills by equipping various kits you find/buy/receive/whatever and that is how your character gets his skills. You modify how well they work by putting your points into the various skills your char has that encompasses them.
Also, your toon learns several skills on his/her own depending on class as you level up.