If your husband works in a state other than your home state, he is required to file a nonresident state income tax return in that other state. He will only report income sourced to the other state on his nonresident state income tax return.
So for example, if your husband travel's to AZ and does work there, he will need to file a nonresident AZ state income tax return and report all AZ source income on this AZ state return.
For your resident state (UT), you will need to report all income (including AZ source income) from sources both inside and outside the state. You will get a state tax credit in UT for the taxes you paid to your nonresident state (AZ) on AZ source income that is also being taxed in UT.
You will want to work on your non-resident AZ state tax return first. You will then take a tax credit from your non-resident AZ return on your resident UT state tax return. (Please note that you will only get a tax credit for your AZ taxes up to the amount of UT taxes that would have been paid if the income was earned in UT).
Just follow the TurboTax guide when working on your states (remembering to do your non-resident state return first) and TurboTax will do all the calculations for the credit to your resident states return
Here is additional information about filing in multiple states (select "see more answer" to view the entire attachment)