You can use both "advantages" and "primal tactics" to get your pet to increase in rank.
Go to your character/persona screen and click on "Warders", you'll see a gauge/bar at the top that says "Journeyman", when that bar reaches a certain percentage it will increase the warder's rank (i.e. Journeyman to Adept). You just need to kill a whole lot to get it to increase.
Start attacking mobs and the weakness symbol will appear (a big blue paw on your screen), when it does you will need to use one of your "advantages". Just keep doing that and it will eventually the warder's rank will increase.