Is there a way to have Corporate Portal accounts go directly to the customer's portal, and they don't see the main website?

When creating the portal there is a setting to set it as a "Corporate Portal". When you do this the customer will receive a link that goes directly to the portal itself.
See screen shot below of where you can find the "Corporate Portal" selection when editing the Portal in "Builder Mode."

If this was not done when creating the portal you would have to delete the portal and restart the process for a portal.