It's *possible* you are confusing yourself more than is necessary. Let's take a step back.
Don't worry about MySQL and phpmyadmin... you were on the right track! You unzipped the Full Package file that you downloaded from this website. You then tried to upload the whole folder into your public_html directory. But that failed. Sounded as though the permissions were not set right for that directory, which is a server issue. Did you contact them?
Assuming you got that worked out, proceed uploading those files.
The next step is to create a database. If you use CPanel with your server, then it's quite easy. But again, that's a question that is best suited for your server techs.
Once all that is out of the way, the easy part really begins. Installation is a snap! Let us know where you are in the process and what, specifically, you're hung up on, and it'll help us be a little more specific in our advice.