OP - Every state I would guess is different. I know in TX, after moving from CT, I had to first do a change of address with the USPS. Once in TX I had to get change my insurance policies to reflect living in TX. I then had to get the vehicles inspects, then go to the tax office to register them. I had to bring proof that I am now living in TX. If you bought a house the closing paperwork of a few pieces of mail addressed to you would work. If renting something from the property management or again some mail would work. In my case I am in a class A RV so I had the park I am staying at write me a letter stating am living their as I am renting a space form them. Once the vehicles were registered I was then able to get a TX drivers licence (from the Texas Department of Public Safety). They want you vehicles registered and insured first. I registered to vote while getting my driver's license.
You need to contact any credit cards and any other accounts and do an address change. Most of which can be done online.
Sounds like a lot but was easy.
Once you have a TX license you can then go and get a Licence to Carry. (requires a class)
I think you know how it is in CT. Go to DMV, first they make sure you owe no taxes, then they talk to you.