<p>I haven't used Visual Studio since VB 3 and am trying to give it a shot with ASP.NET. It seems that it should be able to connect to a website (via some sort of ftp like protocol I figure) and allow to edit without having to manually upload/download the files. Is this the way it is supposed to work or am I mis-understanding? I have tried using 'create new website' and 'open website' using my testing domain (hosted by godaddy, wondering if that may be the issue as well), each time it gives me errors. I'm not sure if I'm doing something wrong or trying to do something it wasn't meant to.</p>

You really don't want to be working directly on a live web site, do you?  That's just crazy.  One little mistake and you've hosed the site.
Visual Studio now has it's own built in web server.  You use that for testing.  If you really don't want to use that you can put IIS on your local machine or set up a Dev/QA server somewhere.  In that case, you'd edit it via a file share.
You should be using some kind of source control.  Even for a single developer it's very important.  When finished with a programming session, you check your updates back into source control.
Finally, only after the site's gone through a suitable QA process, the production server is updated from source control, not from within visual studio.