Install GetSimple on your new/public server (don't worry about username, password, and settings – they will be replaced).

Upload from your localhost/old site the following files and folders (with their contents), overwriting existing ones:

/data/
/plugins/
/theme/

(If your site is not in English, also upload your language file(s) to admin/lang folder.) Now log in again, but with the username and password you used for localhost/old site.

Check the files gsconfig.php and .htaccess at the new destination and compare them with the corresponding ones at the original place. Compare paths, salts, rewrite base etc. Do the edits if necessary.

Go to Setup → Website Base URL:

Change http://localhost/ (or whatever) to your new site URL. Hint: You can copy it from the “Our suggestion is:…” message that should be displayed below.

Click Save Settings

Note: If you have links or images in your pages with complete URLs you should change them to your new URL or else the new site will possibly have broken links or images that cannot be not displayed. Using relative URLs is always advised.