Welcome to the forums.
Install the same operating system as you had on the old server, make sure the hostname is the same and DNS A & MX records point to the new server. Install the same release of Zimbra as you had on the old server then shutdown Zimbra and copy the old /opt/zimbra directory over the new install. You should run the zmfixperms script (as root) that's in /opt/zimbra/libexec and you may need to reset the
LDAP & MySQL passwords. When you've done all that you should install zimbra again (the same version you originally installed), that should bring your system up OK.
Let us know how you get on with that.