WordPress has a system for exporting your data into an XML file at one system, and importing it into another.

By "free hosted" you probably mean wordpress.com. I'm not sure if they've disabled that feature to, perhaps, extort money for export as a $ervice. If the feature is there, it's in the Dashboard under the Tools menu. If not, you'll have to contact them about it.

The XML file will take care of your data--posts, pages, comments, custom fields, categories, and tags--but you'll still need to transfer your attachment files. That's just brute-force upload and download. Knowing how you manage your data, I'll bet you don't have anything on the current site you don't also have on your home PC, so it's just an upload per file. But here's the heartbreaking part: All your image and etc. files will have new URLs based on the date of uploading to the new site, and references to them in your posts, with URLs from the old site, will be broken.

If you can get a list of the URLs to your attachment files on the current site, you could edit the XML file to search and replace the old URLs with new ones.

Or, if it's just a few to fix, you could just do it post-by-post after importing.

Oh, but wait: Your current place is free, so why not just keep it to host your attachment files? The transferred posts will still be pointing back there, new posts will have new URLs...problem solved!