Keeping your mailbox safe
The handful of things that protect your mailbox, and where each one is set.
Most mailboxes are compromised through a weak or reused password rather than anything sophisticated. The steps below deal with that, in the order worth doing them.
Use a strong password
Length matters more than punctuation, and reuse is what turns somebody else's breach into your problem. See choosing a strong password.
To change yours, open Preferences, then General, then under Sign in choose Change Password. If the option is not there, your administrator has disabled it and can change it for you.
Set a recovery address
Add an address you can reach from somewhere other than this mailbox, under Preferences, then Accounts. Zimbra sends a reset code there when you use the Forgot Password link, so without one you depend on your administrator to get back in.
Turn on two-step authentication
Two-step authentication stops a stolen password being enough on its own, which is the single biggest improvement available to you. See setting up two-step authentication.
Some mail programs cannot prompt for a code and will stop signing in once it is on. Give each of those its own passcode rather than turning the feature off again: see create application specific passwords.