Moving your website from one host to another host is fairly simple and for most sites, can be done without any downtime on your website.

 Checklist For Websites Without Databases

Signup with your new hosting provider and purchase a hosting account

FTP your website to the new hosting account. You will be able to check that it works as expected by surfing to the IP address (type the IP address into the browser instead of a domain name that your new host gives you in your Welcome Letter)

Login to your domain name account at your registrar and change the nameservers over to your new hosting company. Typically 2 nameservers are setup - the look like this:
NS1.YOURWEBHOST.COM
NS2.YOURWEBHOST.COM
If you bought your domain from your current hosting company then you may have to ask them to do it.

It normally takes 24-48hrs for the nameservers to update, once they do your website will automatically be served up from your new hosting company

Cancel your old hosting account once the nameservers have updated. A easy way to tell if they have updated is to make a slight change on the old website, once you don't see that anymore then you know it has switched over and you can cancel your old account.

Checklist For Websites With A Databases

If you have a database then you need to ensure that the database retains its integrity and no data is lost in the transfer. If you have a database that is not written too much then the switch can be close to the above scenario.

Signup with your new hosting provider and purchase a hosting account

Backup your database and FTP it to your development PC or your home PC

FTP your website to the new hosting account. You will be able to check that it works as expected by surfing to the IP address (type the IP address into the browser instead of a domain name).
Create a database with the same name as your old database and the restore it using the backup file.

Login to your domain name account and change the nameservers over to your new hosting company. Typically 2 nameservers are setup - the look like this:
NS1.YOURWEBHOST.COM
NS2.YOURWEBHOST.COM
If you bought your domain from your current hosting company then you may have to ask them to do it.

It normally takes 24-48hrs for the nameservers to update, once they do your website will automatically be served up from your new hosting company

Ideally you should stop your website until the nameservers update so that the database maintains its integrity. Remember until the site switches over to the new host, anybody using the existing site will write data to the database on the existing site.

One alternative is to setup the old site so that it writes to the database on the new site. Your new host will be able to help you with this.

Cancel your old hosting account once the nameservers have updated. A easy way to tell if they have updated is to make a slight change on the old website, once you don't see that anymore then you know it has switched over and you can cancel your old account.

ALWAYS BACK UP YOUR WORK!