I don't understand companion missions at all. I have heard of companions gathering while the player is gathering.
I don't understand how to make him go on a mission, I trained him for 2 gathering skills but I was on the space station still (I imagine thats why I coudln't figure out how to send him out gathering) But how can you lose money? Does it cost money to send him out?
Yes, each mission you send your companion on costs money. In beta, the early missions cost 95 to 135 credits IIRC. To send your pet on a mission, click on the crew pane icon (the gem looking one on the top of your UI). From there, you should be able to figure it out.