Go to The Capital City Center, head west to 2nd area (where tavern is located), there is another building on that area, enter it (barrack). Talk to the guard behind the desk (he has “!” on his head). He will ask you about contract, choose the option that let you create it.
Go to the world map and head to Desert Outpost: Fugitive Hideout (Lv 7). At the end of this dungeon, you will see an event. After the battle ends, talk to him and choose 1st option (“?” option; like shown on the screenshot above). He won’t teach you at this moment (Never choose “red face” or “red sword” option).
Go to world map and keep walking until you find encounter that says “Ambush Site”. Offer a help to him, after the battle ends, ask him to teach you (“yellow face”). He will agree and now you can learn about battle mage skill at the Capital City Center Academy.
For battlemage skill, visit THIS PAGE.