If you are pregnant now, and have not yet given birth, you can move wherever you wish to move. Once you have the child, jurisdiction attaches to the state where you give birth. If you move from that jurisdiction without the father's written permission or a court order, you risk having the father object to your move and force the child back to the birth jurisdiction (or possibly even to his custody, depending on the circumstances). He does have more rights if he is on the birth certificate, as that can only happen with an acknowlegment of paternity which means he is accepting the child as his own and indicating he wishes to be a father to that child. Giving the child his last name is an additional piece of the puzzle. Child support is somewhat irrelevant at this stage. You really need to be having a conversation with this man about what he wants, to determine if you have areas of conflict. You also should be having a conversation with an attorney, as there is so much more that goes in to all of this and you have some planning to do that right now you are doing completely blind.
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Legally, you will need permission when a court order establishing his paternity and parental rights are decreed. Once that is in place, you can still move with the child but you risk a court order for return of the child on the grounds that you have interfered with the exercise of his parental rights. Morally, you need to remember always that a child deserves to grow up in the love and guidance of both parents. While the courts will not force the child's father to bond and pursue that relationship, nor even to maintain contact, it is not the right of the mother to deny it. This is a matter of the best interests of the child, and if it is not coming naturally to you, counseling for you could be helpful.
Best wishes for an outcome that serves the best interests of your child, and please remember to designate a best answer.
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Ms. Whitbeck is correct here. The state of birth will control where the case will be had, but you should be sitting down with an experienced child custody attorney to map out your course of action. This is a minefield, and any missteps can cost you dearly. One thing that will help you is if he acknowledges and consents to your relocation prior to birth, though that's ultimately not required. But once the child is born, you are pretty much stuck to litigate paternity and custody in that state, and you will have to get permission of that court to relocate unless the father consents in advance. Again, this is too complex an issue to really go over it here, so I strongly recommend you consult with an attorney before making any significant decisions.