hi,
This sounds like maybe the host is blocking the outgoing mail.
Depending on your server, if it's a dedicated, check if you have WHM, then in WHM try allowing Relay mail in Exim.
It's always a good idea to setup an SPF record though, the server address would be your domain, and any other domain that you want to allow relays from or to send emails using your servers mail transport.
It's been a while since I configured a server for emails, but that's about the gist of it.
Hope this helps