This assumes a straigthforward website on a host; tons of scripts and complicated setups will be more involved. This is a seamless way to transfer your site to a new host, without any downtime.

Step 1: New host

Sign up for an account with your new hosting company, but don't cancel your old host yet.

Step 2: Download files

Use Filezilla to FTP your entire website files to your desktop. A better and easier option is to use CPanel, if your host provides it, to do a full backup of your entire site, which will then be placed on your main hosting account and you can just FTP it to your local desktop.

Step 3: MySQL

If you have any MySQL databases, you can also use CPanel's full backup feature to download these, or use PHPMyAdmin to download them.

Step 4: Upload

Upload all your files to your new host using FTP. Then ask your new host what the URLs of their name servers are.

Step 5: Point nameservers

Log into the website where you bought your domain name from, and change the name servers to the ones you previously got from your new host.

Step 6: Wait

Wait about 48 hours

Step 7: Cancel

Cancel your old hosting account.