If I multi class between a preparation caster and a spontaneous caster, do their spells per day stack?
They exist in parallel, which is not quite the same thing as stacking.
If you are a sorcerer/wizard, say, you get all of your sorcerer spells per day, and you get all of your wizard spells per day, including bonus sorcerer spells for high Charisma and bonus wizard spells for high Intelligence.
However, they don’t get added together, which is to say they don’t really stack. They remain separate, and you have to track them separately. You cannot use sorcerer slots to prepare wizard spells, and you cannot use wizard slots to cast sorcerer spells.