I had an issue trying to forward mail from info@mydomain.com to another email hosted on gmail, and I solved it by setting the MX records on mydomain DNS settings.

How to do this:

After logging into your account, and after you have setup Email Forwarding in Godaddy, go to the My Domains page on the Godaddy website.

In the My Domains page, locate your domain name (e.g. mydomain.com) in the list of domains and click on the 3 dots next to the domain name. Select the Manage DNS option. This will take you to the DNS Management page for your domain (You can also access the DNS Management page through the Domain Settings page and click on the Manage DNS link at the bottom of the page).

On the DNS Management page you will see a list of different Records.  Click the ADD button at the bottom of the Records section.

A dialogue window will open, and the Type field will display. select MX from the Type dropdown list. This will display more fields to complete.

Add the following record:

For the Type field:             MX

For the Host field:             @

For the Points To field:     mailstore1.secureserver.net

For the Priority field:        10

For the TTL field:              1 hour

Then click save. And the dialogue window will close

Add another record by clicking the ADD button again.

Add the following:

For the Type field:                MX

For the Host field:                @

For the Points to field:         smtp.secureserver.net

For the Priority field:            0

TTL:                                       1 hour

Click save and the dialogue window will close.

Give it a few minutes ,and then send an email to your new domain email address(e.g. info@mydomain.com) from another email address (not the email address you are expecting to receive the forwarded messages), and you should now see that the message you sent is forwarded to the email address you wanted to receive the forwarded messages.

Hope this helps someone