Seems like your "regular hosting" uses Apaches mod_userdir, to serve files. This meens, you have a folder beneath your users' home (or whatever is configured), called "public_html".
If you want to switch over to your dedicated server, you have to install a webserver first, you can't just create a "public_html" folder and the files get served!
If you installed a webserver, your HTML-files are going to the configured document_root of your website. If you use Apache, this is by default /var/www/ (Debian/Ubuntu).
Every folder you create beneath /var/www will be accessible via "yourdomain.com/folder".
Your Domain A-Record has to point to the new server of course! To achive this, your provider usually has an interface, to configure DNS. If you can not configure your DNS by yourself, ask your hosting provider for help.