Hi @ooo:
Assuming you want to keep the same domain, it’s really quite simple. Basically you copy the files using an FTP client by downloading from the old host and uploading to the new one, and then you copy the MySQL database by doing a database dump to a SQL script (this is your “export”) and then running the script (and this is your “import”.)
The rest are just details:
1.) Use FTP to download all the files from the web root and all its subdirectories of your old hosted web server.
2.) Upload all those same files to the web root of the new hosted web server in the exact same named directories.
3.) Do a full database dump (i.e. export) of your MySQL database as a SQL script.
4.) Create a new MySQL database at your new web host.
5.) Create a user for your MySQL database and set it to have all permissions. Be sure to record the name of the database, of the user and of the password.
6.) Import your MySQL database by running the MySQL script in context of your new database.
7.) Next edit your /wp-config.php file on your new server and give set DB_NAME, DB_USER and DB_PASSWORD to your new database name, new database user name and new database user password.
7.) Finally, be sure to change your domain’s DNS settings so it uses the name servers from your new web host. You;ll need to wait the timeout for the DNS cache to clear and the addresses to propagate (typically it only takes a few hours but it can take up to 48 before your new site is accessible.)
Thats about it. If you need more specifics be sure to give a lot more details about your new host including minimally the name and website of your new host and if possible what type of control panel they offer for you to administer your site (i.e. CPanel or something else?)