WordPress is one of the most popular content management systems (CMS) in the world, and as such, many web hosting providers offer options that cater to this rather large niche. You can certainly host your WordPress site on a standard shared hosting plan, but by choosing a WordPress-specific option, you'll spend less time configuring and managing your WordPress installation.
Moving to another website consists of transferring the website’s files and databases, configuring your site with the new host, and directing your domain’s DNS to the new host. Once you pick a new site host, they can usually help you out with this process. The cost will depend on the host you’re switching to, but it will probably be anywhere from $150-$400.