When they are due.. pay them. There is no tax exempt period. It begins the day you move there.

You must file by April 15 in your new state, just as with federal taxes. For the first year you are most likley going to file as a part-year resident of your new state, and you pay tax to the new state only on income you earned while either working or residing in the state. You also file as a part-year resient of the old state, to cover income earned while working or residing there.

If your question is about paying estimated income tax - the first year you probably can get away without having to pay estimated tax without any interest penalty, as most states (maybe all? Not sure) allow an exemption against interest penalties for underpayment if you previously had no taxes due to that state. But this mught mean you have a pretty big tax bill due in April. Starting the second year you should pay quarterly estimated taxes if you owe them, or you run the risk of facing a late payment interest charge.