Exporting and Importing from one Zimbra account to another is very simple:
Import/Export lets you choose what data to export (Account, Calendar, or Contacts) and from which folder. You can also use the Advanced Settings checkbox for even more granular searching.
In order to include messages in your Spam and/or Trash folders in your export, you'll need to click Preferences then General and check Include Spam folder in searches and/or Include Trash folder in searches first.
Importing from other services is also possible, the PST Import Wizard is available to help import from PST (Outlook) files and the Zimbra Outlook Connector is available for Premium Edition users as well.
Other messaging services can usually be migrated simply as well via. HowToForge has over IMAPSync as well.