If you're a newbie of Linux, you'd probably be better off using the OS you are familiar with to run your DNS server. You really shouldn't run DNS on the same server as your Zimbra server anyway, unless you have to, and that looks like exactly what you are trying to do. That said, your DNS setup may not be the reason you can't send and receive email in the first place.
For a good primer on BIND9, check out this page:
https://help.ubuntu.com/community/BIND9ServerHowto
A forum like this is really not the best place for someone to teach you BIND from scratch. Use that link as a primer, and my original like for a more in-depth study.
That said, my Silly Wild Academic Guess would be that you haven't indicated any forwarders in the named.conf.options file, which would prevent the Zimbra server from resolving other Internet domain names, thus preventing it from sending mail out. And, if you are trying to receive email FROM the Internet, then you need a domain and a public DNS entry for the MX record (and a public A record) for your Zimbra server that works from the Internet. And that's a whole other discussion (public facing IPs, NAT forwarding, dynamic DNS and such).
Again, these are things that are best learned as part of a general education on DNS.