That you're using Cyberduck FTP is irrelevant. Any FTP client will do (FileZilla is probably the most popular). Note that some directories or files may be protected (i.e., Read-Only) and you need to know how to unprotect them before erasing them (as well as knowing whether they're system and should be kept, or something you put there and can safely delete).
The first thing to do is to make a complete backup of your site's files (and any databases, too) and download it to your PC, so you have a copy of everything. Make sure you know how to restore everything, or just selected items. It sounds like you don't want to keep anything from your old site, but as soon as you erase everything, you'll remember that you wanted to keep a certain useful piece of code...
If you have any public FTP capabilities (letting others upload files to your site), you want to use your control panel to remove/reset all those FTP accounts (if you don't want them to continue to be used). Erase all the uploaded files. Look at your control panel to see if you have "password protection" on any directories that you plan to get rid of, and properly remove those protections before erasing the files and directories. That way you won't be left with "dangling" protection entries that point to nothing, and could get your server upset. Use your mail system to erase any mail entries (accounts) and emails that you no longer want to keep, rather than blindly erasing files in the mail system. In general, anything that looks "systemish" you can either leave alone, or use the control panel to properly remove it, rather than risk mangling your system by erasing vital things.
Everything else of concern (publicly accessible) is under public_html/. Leave any directories that you don't recall creating for your purposes (such as cgi-bin/ and scgi-bin/), and erase everything else in public_html/ and below. Make sure that you have an .htaccess file (even an empty one), if the server doesn't re-create it for you. If in doubt, keep the old .htaccess and only edit out things you know you (or an application) added. There could be important system settings in there. If you have installed various applications for your old site, they may have created their own directories (folders) in public_html/, or even above that. Your iWeb application may have created some extra stuff, too. That's why you made a backup of your files, so if you find you've erased too much, you can restore it. It's difficult to be any more specific than that, as I don't know the history of your site and what originally came with it. I've got plenty of stuff on mine that was put there by the system, but probably is no longer used and I could get rid of if I wanted to.
Consider what you want to do with visitors who show up looking for your old site, especially if they are coming in to "deep links" to specific old pages. Do you want to direct them to new pages, or just let them "404"?
Finally, any site owner needs to know what's in their site, what it does, and who put it there. That way you can safely make mass changes (as you want to do now) without causing a catastrophe. This time around, keep records of all changes made to your site, so you know these things and don't have to ask what can be removed.