What you Need to Know
Ask your current web administrator to provide you with the FTP username and password. This will provide access to the different pictures and files that make up your website. Some web administrators may only have one FTP user for all of the websites that they maintain and will not be willing to give access to other peoples websites. If this is the case, ask that they send an archive (.zip file) of the website files.
Domains are not copied and pasted from one server to another like the pictures and files are. If your old web administrator holds your domain name you will need to ask them to unlock it and send you the authorization code (or EPP). The domain will then be ready to transfer to another registrar, either into your control or to the new web administrator. The domain does not need to be transferred if you registered it and can make changes. The DNS server will need to be updated to reflect the server change. If you do not know how to update the DNS, ask your web administrator.
Send the FTP login details and the domain authorization code (if transferring your domain) to the new web administrator and they will move your website. A email will be sent to the The admin contact listed in the Whois database asking for authorization to move the domain. After permission is granted the domain will be transferred and the website will be up and running again shortly.
Not Familiar with the Whois Database?
To see who is authorized to make changes to your domain you can search the Whois Database. There will be three different contacts listed: registrant, admin, and tech. The registrant is the owner of the domain, the admin is the contact for administrative, non-technical issues, and the tech is the contact for technical issues. If the domain is your business, your name information should be listed as the registrant and admin. The tech contact will typically be your web administrator, if you have one.