It is not recommended to cancel your old web hosting account until the transition to the new web hosting company is complete.

Step 1: If you don't have the files backed up on your hard drive already then using a basic FTP program transfer all your website files and pages. Remember to keep the same directory structure.

Step 2: Find your new web hosting provider.

Step 3: Register and purchase your new hosting account. If you already have a domain name be sure to select that you will keep your existing domain.

Step 4: The DNS of your new web host must be updated in your domain. This information will be given to you by your new web hosting company. Login to your account with your domain registrar. Your registrar is the company where you initially bought your domain name from unless you have transferred it elsewhere. If your domain's registrar is your old host than just contact their customer support and ask for the name server information for your domain. After gathering the new name server information change it on the domain's settings.

Step 5: If your new host provides you with temporary login information try to upload your website before the DNS values resolves, this way when your domain points to the new server your website won't be down. It can take DNS about 48 hours to propogate. People will not be able to see your website during this process.