No, you'll just provide them with the new address.
Will I be able to register it in my new state?
Of course you will.. Nothing changes.. people move all the time and don't have to change lenders or cars ... Just inform them of new address, change insurance if needed and register it in new state along with new DL.
Welcome to the myFICO forums!
With online banking, you should be able to continue to use your current CU. With direct deposit and mobile banking, there really isn't a need to change CUs. On the other hand, the CU that my employer is affiliated with is still in the dark ages and has no app or mobile banking. YEMV