When I set up new servers, I like to make sure any system messages like cron failures, server issues, or emails that are routed to
firstname.lastname@example.org (where 'example.com' is the hostname of the server—meaning emails to that domain will get routed through the server itself and not hit an external MX server unless postfix/sendmail is configured correctly) are sent to my own email address.
It's relatively straightforward to route emails to internal users (like
root, etc.) to an external email address; you simply need to edit the
/etc/aliases file, adding a rule like the one below, then run the command
# Person who should get root's mail
By default, most internal users are routed to root as well (including webmaster), so setting an external email address (or a list of addresses, separated by comma) for the root account will allow you to more easily see what's happening on your server. Don't forget to run
sudo newaliases to pick up the changes!