Hi Sami,
This is what I do for Arizona. (Others may have better info.) :
I have a TPT License (Transactional Privilege Tax), which is what Arizona calls its sales tax license. You won't owe tax on items you ship out of state, but you will for items sold in Arizona. But you still have to report all sales. You can get more info and the license at the Arizona Department of Revenue website.
After I signed up, I got a notice from my city (Surprise) that I needed to have a business license from the city. However, I didn't need it because I met one of the exemptions (handcrafted items with sales below the minimum amount.) If you get a similar type of letter, make sure you check out the law. You might not be required to have one. The city gave lots of details on how to get the license but not on the exemptions. Now if my shop takes off and I sell, sell, sell, then I'll gladly get it. :)
Depending on where you live in Arizona, you can use the one TPT form for both state and city tax, if any.
Good luck! The website information isn't as clear as I think it could be. So if anyone else has better info, please let all of us know!