If I understand you correctly, what you're wanting to do is move your databases and websites from your local computer to your paid-for-hosting computer, correct?
I'm guessing you're developing your site locally on your computer, then putting it on your hosting site to make it public.
This can be pretty easy, but it can also be a real pain in the butt too.
I have had success at moving forums as an experiment from one place to another and here's how I do that:
Create a backup of your MySQL backup of that database for that forum. It should create a text document.
Copy the root folder and all of it's contents, including subdirectories for the forum.
Upload the files to the new root folder for the site.
Now go into your new phpMyAdmin and create a new database, probably should be the same name as the old one.
Choose the option to "Import" a database and make sure you have the new one selected on the left.
Find the file and import it. With any luck at all, it should import just fine. If you have large databases, you have to use something like bigdump.php to move the database, which you can get it from http://www.ozerov.de/bigdump.php
. Do a search on how to use it.
For a phpBB forum, this article
may help or not work for you (it didn't for me), but it'll get you on the right track.
One of the biggest problems you will probably encounter is going to be your paths not being correct in your settings. If that's the case, you'll have to work through them one by one until you get it figured out.