Dan,
You are talking about a 'relay' when you want zimbra to send email to another smtp server on the way out. That is set under your 'Global Settings' and 'MTA' tab. You can only set one 'Relay' Server in the GUI for all domains on your zimbra server.
Next issue, most servers these days, mine does, 'black list' email servers on dynamic IP address. Typically smtp servers on a dynamic IP address could be a spammer. Check your IP address and see if its on the 'black' list.
The Spamhaus Project - PBL - The Policy Block List Lookup an IP Address in the Spamhaus DNSBLs
Next thing to watch, some ISPs are intercepting smtp from residential users to help prevent spam. You can test if your ISP is intercepting your smtp traffic by telneting to one of your email accounts smtp server like this: telnet mail.domain.com 25. Use port 25 and use the domain or IP address of your email accounts smtp. If the name of your ISP shows up they are intercepting, if the name of your email server shows up you are good for now.
Question, what would the benefit be by gathering all your email from other email servers onto zimbra first before your email client?