Upgrading a Rails application incrementally

One of the major projects I led at my last Rails gig was upgrading our application from Rails 3.2 to Rails 4.2. I'd like to share some lessons I learned about upgrading a large Rails application with minimal problems. Some operational issues I'll discuss are specific to the Rails 3.2 to 4.2 upgrade, but most of this should be relevant to any incremental Rails upgrade.