How to become my dad's caregiver when my mother is not fit to take care of him but she's his legal his Guardian?

Ryanterry, you would have to petition the probate court for change of guardianship. One would have to be canceled and the other assigned by the court. This will cost money. There is no way around that. You may also have to hire counsel to represent you and to push things through. All this would be paid out of your father's money if your petition is successful. I do hope you can find an easier way to work through this. It is a good question about what to do if a guardian becomes incompetent. In the case of children, the state will step in if a parent is shown to be unfit. I imagine the same type procedure is in place for elders.

BTW, you would have to present evidence that his current guardian is not competent. That would require doctors or other experts to put it in writing.

First you need a few thousand dollars to retain a lawyer.
Second you need hard proof she is doing a bad job.
Third, to be a Guardian yourself, you need a fingerprint and background check and a credit check. No credit or bad credit or even a bad driving record could disqualify you. If you cannot afford a lawyer and you cannot afford to post a bond, it isn't going to happen.

Adult Protective Services may be the only way to deal with this without going to court yourself.