Thanks for providing that info, Emmie! Looking into your account now; Will update you as soon as I have more info for you. Stand by!
So it looks like you've copied your user directory from the old host into the site directory here, based on the way /logs and /public_ftp and /public_html are listed there. In any case, it's always going to display like that until you have an index file like index.html or index.php to cover up the directory listing.
I suggest using Filezilla to move things around properly and clean up your directories. Since I'm not sure exactly what you wanted to keep, I'm thinking you might not need most of what you uploaded except for the public_html directory, which should be made your site directory (maybe move it up a level and then rename). OR you can empty the contents of it into the existing directory it's under now, whichever works for you.