Email forwarders send a copy of emails and keep an original. You can setup multiple forwarders to and/or from the same account.

I have found that if you send email from Gmail to an email address which forwards back to the same Gmail, then Gmail will not receive that forward.
How do I delete the mailbox from my email accounts?

Can I have all the outgoing email forwarded to my email address?
No, this can only be done on a dedicated server.

Is it possible to set up e-mail accounts that forward to more than one e-mail address?
Yes you can. You just need to log into your site's cPanel and click the Forwarders icon, and then add a forwarder for each destination, like so:
This email address is being protected from spambots. You need JavaScript enabled to view it. --> This email address is being protected from spambots. You need JavaScript enabled to view it. (click the Add button)
This email address is being protected from spambots. You need JavaScript enabled to view it. --> This email address is being protected from spambots. You need JavaScript enabled to view it. (click the Add button)
This email address is being protected from spambots. You need JavaScript enabled to view it. --> This email address is being protected from spambots. You need JavaScript enabled to view it. (click the Add button)