Magicka regenerates naturally based on willpower, so higher willpower means faster recovery even during combat until you cast a spell. If you want instant refills, using wait for an hour works perfectly fine. Things get interesting with the atronach birthsign though since it completely blocks natural regeneration, but gives you 150 extra magicka and absorbs half the spells cast at you.

Running with atronach means youll need to plan ahead since natural regen is gone, but you've got options. Welkynd stones are your best friend for instant refills, and you can also use magicka potions or hit up ayleid wells for 400 points each. Intelligence affects your max magicka pool, but your regen speed only cares about willpower, so keep that in mind when leveling. Smart spell absorption and resource management are key here especially early on when supplies might be limited.