In every elder scrolls game, there's always one category in your player's menu to choose which magic you'd like to be currently using, but I can't seem to find it, does anyone know where it is, beside staffs?
The "magic" you're looking for doesn't exist in the same way as past ES games, like Oblivion or Skyrim. Your "magic" skills are that of your class skill lines or from the destor/resto/mage guild, etc skill lines.
You can't really "equip" a type of magic in ESO. Any abilities that use magicka are generally considered to be "spells". A staff isn't required to cast spells, but staff abilities are classified as spells.
Look at your class abilities (I.e. the abilities in the three trees tied your character's class, which is either Dragonknight, Nightblade, Sorcerer, or Templar) and their descriptions to see what kind of damage they do. There's several different damage types you might see, including Magic, Physical, Bleed, Poison, Fire, and Shock, to name a few. Abilities that require magicka to cast will scale with Spell Damage and those that require stamina to cast will scale will Weapon Damage.
This will sadden you greatly, OP, but you will get used to it:
ESO, unlike main ES games, does not really have much concept of what you are wielding with respect to abilities.
For instance, one would imagine that to activate Killer's Blade, one would need to equip a dagger, as it's sort of like a backstab.
However, if you are wielding a greatsword, you can still use this attack. A magical red blade will appear in your hand, and your greatsword goes to wtfland for a moment.
Even more bizarre, is that even if you are wielding daggers, your dagger still will not be used during the animation.
I was very dismayed that the game basically has no real binding between my wielded weapon and my abilities, outside of the restrictions of the skill trees themselves (e.g. you have to dual wield to use dual wield tree abilities).