For contacts you can also use a product called
Zindus
It will sync your contacts between Thunderbird and Zimbra. I don't use it regularly, but have used it in the past.
One thing to watch out with migrating email is that if you setup Zimbra in Thunderbird and just drag the emails in, the timestamps in the web interface will be set to the time that it was copied. Once you open the message it will show up correctly, but in the listing they will be wrong. I would strongly recommend using imapsync if possible.