With Open Source, you have these options for what you want:
PC: Zimbra Web Client, Zimbra Desktop, IMAP/CalDAV client. The most popular example of the latter is Thunderbird + Lightning, but you might hit a few bumps. Also, you'll need a way to sync contacts. CardDAV would be best, but at the moment I think the only option for Thunderbird is Zindus. (There are other IMAP/CalDAV clients; they just don't get as much discussion, so I can't say how well they work currently.)
Mac & Linux have similar options but it doesn't sound like you use those for your workstations/laptop so I won't go into detail. Note the ZWC requires you to have an Internet connection to do anything; ZD or an IMAP/CalDAV client will let you do some things offline.
Phone: if your phone supports IMAP/CalDAV, you can just use those. E.g. iPhone. iPhone with CardDAV may also work--you'll have to search the forum and test to see what the current status is. Many phones support at least IMAP for email. CalDAV may be available for Android, see for example
Calendar (CalDAV) Sync BETA. To fill in the holes, you may be able to get by with the mobile version of ZWC. A specialized option for iPhone is the Aderium Companion for iPhone, which is a third-party (paid) app whose functionality is similar to Zimbra Desktop.