First off, I greatly apologize for the delay in getting back to this post. It slipped off my radar. Hopefully the answers below get you going, but if not, let me know and we'll be much more responsive going forward.
I couldn't find a page that was showing the error you mention when I try to hit the Like button. The only page I actually found the Like button on was /index.php/welcome and it worked there. Running that page through the URL Debugger also didn't show the error you mention, so I'm not sure if I'm looking in the wrong place or what.. let me know if there's a different page to test on.
In your PM, you mentioned:
I really don't need an app in FB (unless you tell me otherwise as maybe I am blinded in some way - which is possible)...yet I would like to have some presence in the social media world to help market the site and program.
You don't need an AppID for the Like button, but it is required for a lot of social stuff like authentication (which looks like it would be beneficial for your site), Facebook invitations for friends, Open Graph Actions and a few other things.
As for recommendations. From what I see on your site, you don't have the Open Graph - Content plugin enabled. You should enable that which will automatically set better Open Graph tags on your Joomla Content pages. That plugin will set the image tag based on the article contents as well as the description. It's better to do that than to let Facebook 'guess' what's important about the page.
I hope that gets you going a little bit more, but if not, let me know.. Again, we'll help how we can.. and faster
Thanks,
Alex