ICANN publishes the complete list of TLD registries and domain name registrars. Registrant information associated with domain names is maintained in an online database accessible with the WHOIS protocol. For most of the 250 country code top-level domains (ccTLDs), the domain registries maintain the WHOIS (Registrant, name servers, expiration dates, etc.) information.
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.