I graduated college in June. I live with dad. Gave birth to a child in April. Dad pays insurance/living expenses. He claimed me/daughter. Do I list my child on 1040?

[xmasbaby0]

No, you do not list your child on your own tax return.  You say you and your child are being claimed as dependents on your father's return.  A dependent cannot have a dependent.  You can file your own tax return to seek a refund of tax withheld from your own earnings, but you need to say in My Info that you can be claimed as a dependent on someone else's tax return.