Backup! Backup! Backup!
The first step in the transferring process is to backup all your files including database files. You can use a good FTP program and download all your files from your old hosting company into your local computer.
Manually Move your Files
Some people will need to manually move their files. Those without a database will need to transfer all of their files to the new server. Those who have a database will simply need to move the database to the new server. Work with your new web host to understand the easiest way to move your files to their server.
Hire Someone to Move Your Files
Some people do not have the time to actually move their files. You may hire someone to move your files for you. Sometimes, the new web host will move your files for you, at a cost. Other companies exist to simply move your files for you. If you know how to do it, move your own files. If you do not know how, you may be better off hiring someone to do so.
Move your Domain
You need to make sure that you keep your domain up to date. Work with the web hosting provider to ensure that your domain works with your new site. You also need to change the information with your domain provider. You simply need to update the web host provider information for the domain.
End your Old Account
Believe it or not, some website owners forget to officially end their old account. They will still be charged for the service until they do. You simply need to end the service with the company. Do not end the service until you already have your website up and running under the new host.
Popular web hosting providers like WebHostingHub do free website transfer. In fact, WebHostingHub is offering a "no-downtime transfer process" which means they ensure that your site will never be down while transferring process happens.