At the beginning of Terrain of Magical Expertise, you will create your character including choosing two special attacks. At the start, you can choose both an elemental attack and a neutral attack. You will have the opportunity to learn new throughout the game.
To learn new special attacks, you must travel to the dojo of each city. There you can challenge the dojo master. If you defeat the dojo master then you can learn a new special attack. You must battle the dojo master alone so it would be good to stock up on items.
Each dojo will offer a selection of special attacks to choose from. Before you make a decision, you can try them out. This lets you battle a dummy with all of the new special attacks at your disposal. You can practice each one's inputs as well as see how much damage they do.
When choosing a new special attack, it is recommended to choose one that compliments your party. Your character and each ally have elemental attacks already, so it would be wise to diversify what kind of attacks you have. You can also choose a neutral attack that has no strengths or weaknesses. As there are multiple dojos, you will be able to get multiple new special attacks.
For more information on Terrain of Magical Expertise, check out Where to find the secret path in the Luteus Labyrinth in Terrain of Magical Expertise and Is there a map in Terrain of Magical Expertise on Pro Game Guides.