You can use the open standards CardDAV and CalDAV to easily synchronise your Zimbra Contacts and Calendars to your Android smartphone.
Android has no native CardDAV client but some are available from the Google PlayStore and F-Droid.org. We now recommend the free DavDroid which works well. We talk you through configuring that client but essentially any others that are available will be similar. To sync Tasks with DavDroid install the Open Tasks app from the store/F-Droid and then re-install DavDroid.
Once you install DavDroid it will appear in the Android Account Sync area, but simply starting DavDroid will give you the option to Go to accounts & Sync.
Select Add Account, and choose Login with URL and username as the account type.
The server name is eg https://zimbra.opensure.net.
Enter your email address and password.
The app will then check the connection and ask for an account name for you to identify the configuration by, eg the email address. Change this as or if desired. Leave the groups as separate VCards.
Your contacts and calendar (and Tasks) are now able to sync.
To change which calendars and address books will sync, open the Accounts area under Android Settings and click on the DavDroid section listing the DavDroid accounts. Select Manage Accounts and Click on the account shown that you want to configure. Select the Calendars and Contacts from the list that you want to sync to/from your phone.
The Settings Cog under each accounts area will allow you to more finely configure things like intervals, connections etc.