For the most part, you don't. Your sovereign and champs will start with certain spellcasting affinities, according to the different shard types: fire, water, life, death, air, earth. As you level up you will be given the opportunity to upgrade your spellcaster's skill at casting those affinities. Each time you select, for example, a fire magic trait, you'll get access to more fire spells. You can't pick up spells from a school you don't have an affinity for to start with. So, if you find a champ that has no affinities whatsoever, they will never be able learn them.

Spells that are unlocked by research do not need any hero traits to cast in the current beta version. Spells that are unlocked by research can be found in the Magic tech tree, under various techs. Exactly what spells appear as researchable change from game to game.

So you need to level up your character to automatically gain new spells?