Moving a website is a pretty simple process.
Here is a short instuction of how to transfer website to new host:
1. You move the website's files (you can create a backup of your site's files on your current hosting with web-interface and then copy it to a new one OR you can do the same with any FTP manager).
2. Move the website's databases (usual, hosting provides such opportunity through web-interface; if no, then you can make it directly through phpmyadmin by exporting your databases on one hosting and import them on another).
3. Point your domain's DNS to the new host (you probably already know this part since you have your site working on current hosting).