Hi Arthur,
You are correct, ALL files in public_html is your site's file!
First step, you don't have to "transfert" your domain, as you don't
want to keep the "host22.com" domain ("altonaweather" was
a subdomain of host22. and subdomain aren't registred)... then
doesn't need to be transfered. You need to "buy" your domain
name (ex. altonaweather.com), this will give you an ftp access
to your site (ex. ftp.altonaweather.com)
Second step: As I told you, you have to take all files contained
in public_html of your current site and transfer them to your Pc.
Third step, you have to take inverse "road"; take all you copied
files in step 2 (on your Pc) and send them with your ftp program
to your new domain's public_html folder.
Finally, your site will exist on 2 places. then you could close
host22 account, or keep it to redirect visitors to newer site!
Regards,
Jacques