Can I move out of state with my daughter without telling her father since there's no court order

If there is no court ordered visitation or custody, then you can move. However, you can't hide from the father. If you don't tell the father where you are, you could find yourself charged with parental kidnapping.

And of course the father can try to file for custody and/or visits ( assume you live in the same state now) in his state, before you have gotten residency in your new state.

But you can move, but then he may have options in court also if he desires to try to use them

Yes, you can move but you need to inform him where to send the child support paymens to and he can take it from there to figure out how to work out visitation (on his dime, not yours).

If no visitation schedule was set by the court then it would be up to him to file for visitation.

The child support would not be affected and you would not have to go to court to have him continue paying since you moved if it had been court ordered.