Hi! From the Player's Guide page on Training Companions: "Like your pirate, Companions can be trained. Every time you level up, you’ll also gain Companion Training points that you can use to train up any of your Companions. You can see how many total unused training points you have, by looking just to the left of the Train button on the bottom right of the Companion Management screen. Training does not cost anything except Training points."
If you're wondering about Companion promotions, I think you'll also find this information in the Player's Guide handy.