What are some important security-related error codes / events to monitor?
It's a good idea to create SQL alerts in the SQL agent for all errors of severity 16 through to 25. Also create 1 SQL alert for each of these error IDs 823 824 825.

I would create an operator that points to an AD group (probably DBA or ITsupport) so that a team can be alerted or you can drop users in and out of that group when you are away so that errors are not missed. 

This will keep you in the loop when things are happening on your server that you should know about but something like Redgate SQL Monitor is a much better solution.