Should I let my mother's mental illness make me move out to my dad's house
Have you discussed this to her?
No, she's in the hospital.
You should respect your parents but depending on the severity of her illness your mother might not be in a position to make good decisions right now. You should talk to your father about this.
No, I already plan to move out to my dad's house.
You really need to speak with at least one of your parents to make sure this move is okay with them. I'm sure they can work with you to figure out the best place for you to live.