In the character bubble-the one on the far left of the line of bubbles on the bottom right of your screen-go to companion, then the skills tab on the top. There, you can allocate points for whatever class type you want. As for leveling, it just takes a lot of time and experience for your bird. Fate farming is the best way. You can ALSO improve him/her with the use of a house stable and the carrots (I can't remember what they're called.) This is a slower method, but very certain. You have a challenge log for your bird (in the "battle" tab) and it helps immensely, but again fates will accomplish these as well.