Check your page permissions, by opening the sitemap left clicking a page, and than clicking permissions. Make sure read is available otherwise they will get a login.

The other thing you want to check is system & settings to make sure the viewing permissions aren't set to private.

These are great ideas however, I can't get to the sitemap or the dashboard. I get the login page no matter what page I try to pull up, even if I log in I still get the login form. I am logged in at that point because Sign out link is shown and the edit bar appears at the top of the page.

Does the "sign out" link continue to appear if you refresh the page? If it doesn't, that might indicate that you have a cookie problem.

If logged in as admin, you should always be able to get to the dashboard using /index.php/dashboard.

If you have turned off pretty urls, you might need to adjust your .htaccess file too. Maybe there's something else in there that's incompatible with your new environment. It might be worth renaming it ut of the way to rule it out.

Also check the contents of config/site.php. If your BASE_URL is still set to your old site, you're probably being redirected back to it and not actually logging in to the site you think you are.