Can you set up responses to go to a certain email address? Or even just a notification that there is a new response?

These are our guides on setting up email notifications.  The first is on setting up notifications sent to you:

http://www.jotform.com/help/25-Setting-Up-Email-Notifications

The second is how to setup notifications to your users:

http://www.jotform.com/help/26-Creating-a-Form-Autoresponder