Skylab, what exactly do you mean by "SharePoint website database." Do you just mean SharePoint itself? When you use InfoPath with SharePoint, it has to be published directly to a form library, and that is where the users go to fill out the form by clicking New. This is also where the form will be submitted upon completion and therefore where all your data will reside. Each form gets saved as a separate XML file, but the fields of the form are saved into the form library as columns as long as you promote them during the InfoPath publishing process...just as Ted stated. Don't think of it as submitting to a database. Everything in SharePoint goes into the content database, which is in SQL server, but it's not handled like a normal DB. Your data will just go to the form library, which is where you will see/review it.
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Connect "both" what?
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In the form library where the form is published and submitted
When you say "visitors," do you mean people who aren't users of your system? If so, then InfoPath doesn't work for this unless we're talking about SharePoint 2010 list forms. Anonymous users cannot submit documents to SharePoint, which includes InfoPath forms. The methods explained here will only work for authenticated users.