“We have moved from Florida to New Mexico and want to start a permaculture design process. How would that look? What is the first thing we should do?”
(per Mollison) The first step! 1. *Close, protracted* OBSERVATION of the site. (Ref. Introduction to Permaculture, book by Bill Mollison with Reny Slay.)
And in the OBSERVATION, include the social landscape of your new place. Who are the nearby permie people and groups? mutual aid networks etc. And of course garden groups, forager groups, any nearby food forests, community meetups etc. Observe closely so you can tie into existing networks/energies and not reinvent the wheel. And also because your wonderful skills & energies, experiential capital, and social capital will surely be much needed & appreciated as new additions to the local ecosystems!
It’s great that you started going hiking around your new home right away. I bet you made many useful observations.
And, on your site right now immediately: start collecting water (condensation off the roof; condensation on rocks & metal tubs; of course and rainfall etc.) & putting it on the landscape. Collect greywater and put it outside on the land too; try not to let any water go down the drain.