1) Make a dump (backup) of your database.
2) FTP the code from your current host to your new host.
3) Create a database on your new host. Make certain that the database name and username is the same as it is on the current platform. If this is not possible, then you'll need to modify your code to suit the new platform.
4) Execute the SQL script created in step 1 to restore your database.
5) Your new host should provide you with a temporary URL that you can use for testing. Test the site thoroughly.
6) After thoroughly testing the site, login to the domain registrar and change your name servers (DNS).
*NB: Don't forget to consider email (if applicable).