Regarding day-to-day housekeeping of Linux systems I'd consider log monitoring one of the most important things to do - not just as a sysadmin.
Of course you don't want to do this manually but rather use tools like logcheck for this task. They watch system logs, check entries agains a whitelist and mail you all the lines that don't match, i.e. may indicate issues (e.g. hardware failures, security breaches, etc.).
For me this has worked for many years but when it came to systemd and volatile logs via journald it didn't anymore.
Enter a new project: journalcheck
journalcheck uses journalctl to grab system logs and egrep to
match against a set of whitelist filters provided as plain files. It's
written in Bash, very simple but gets the job done.