Moving to Spain from USA

Hello! My husband is a citizen of Italy, Guatemala, and the U.S. I am a U.S. citizen and we have two U.S. citizen-born children.

We would like to move to an EU country (right now we are looking at either Spain or Italy, but most likely Spain). What is the fastest way to legally immigrate to the EU?

From my research it seems like we could enter the EU (my husband as a citizen, myself and the children as tourists). My husband would register as living there and then myself and the children could apply for residency.

That seems almost too easy, so I wanted to see if anyone else is a clear understanding of the process. Thank you!