You need to move a copy of the entire site—files, database and all—across to the new server. Does either server have CPanel? If so, you can do an automatic backup of everything in one hit and move it all over. Often your new web host will help to move everything over.
At least 24 hours before you move the site I suggest asking your current host to lower your TTL to 300. This will resolve traffic to the new host 5 minutes after changing the name servers.