Before attempting any type of site move or upgrade, it is important to backup your site - both files and the database.
Backing up your files
To backup files, you can access your site using FTP, or using your hosting account's control panel, which should have some type of file manager or webftp option.
All uploads in WordPress end up in a folder called "uploads" which can be found in the "wp-content" directory. Simply download this folder and you will have a copy of all your site's uploads.
If you have been blogging for awhile, this folder can be very large and may take a long time to download.
Backing Up Your Database
Backing up a database can be a little trickier.
The easiest way is to use a WordPress plugin called "WP-DBManager" to easily backup your database and email it to yourself. More about WP-DBManager.
You can also use something called "PHPMyAdmin" if your current host makes that available.
You hosting account's support may be able to help with this if you totally stuck.