Should I download/start my resident state or nonresident state software first?

Start and complete the non-resident tax return before you start your resident tax return.  This is so that you will get a tax credit on your resident return for the taxes paid on the non-resident tax return.

Prepare the non-resident state first.

https://ttlc.intuit.com/questions/2895920-why-would-i-have-to-file-a-nonresident-state-return