Hello, I am a mother of a three month old baby. I am in a bad relationship with the baby's
Father. Do I have the right to take her with me until custody is established or am I expected to leave her in the current residence with her father?

Sure you can take her but until custody is established he has equal rights to come and take her. You need to for custody ASAP so find a lawyer in the morning (if you haven't already). If it is an abusive relationship you can always go to a shelter and they have protection for you and your child.