The first and most frequent thing I check is emails. Locally we get emails for job failures, deadlocks, long running processes, tempdb growth, and a few other health checks. The majority of our databases have associated dev/qa environments that get refreshed from a combination of full/diff/log backups every day and I get notified if something goes sideways on that. Other databases are not critical at all and get tested infrequently. They all also get synced over to a DR site that's always live.

I've got a number of auditing jobs that reach out to our servers and check for various failures like job and database mail failures. They also populate and update a central DBA database that houses inventory information (instance level, database level, server level), job run history for all the instances, database file information history, server drive information history, etc... Even though we have failure notifications set up locally on jobs, sometimes who gets notified gets changed without my seeing it - my auditing notifies me of all job failures on any of our instances whether I'm set up to get notified or not. It also notifies me when security changes are made on the servers and if there are connection failures while attempting to audit anything. At the start and end of every day I get a summary email from the central DBA database of various failures, security changes, and an overview of how much each database grew in the last 24 hours. 

Most of our resource monitoring is done in WhatsUp Gold so I have a few dashboards set up there so I can just watch things like memory and cpu and disk throughout the day without thinking about it.