If your "new" website doesn't use the 'old' files, then why not just delete them?
First, save them all on your PC (for archive in case you need to see them), and remove them from the website.
If there are still links out on the internet to the 'old' files, create a 404 page that redirects users to your new site main page (404 web page does not exist error). Most webhosts have a way to control error pages.
Yes, use FTP to manage files between your computer and website. FileZilla is a free FTP client. FTP accounts are managed via your website control panel (on the webhost account).