Take a copy of the files and a copy of the database before you change a name server. The copy should live somewhere other than the old host, because the old host may close the account when you leave.
On the new plan, create an empty site, import the database, and upload the files. Update the site address inside the application if it stores one. For WordPress that address is in the same database you just imported. A search on the old hostname is safer than a blind replace if the hostname also appears in serialised fields.
Only after a test on a temporary address, or after you have checked the site by editing your own computer's hosts file, should you change public name servers. Keep the old host until the 48-hour window has passed and a person who is not you can load the site.
The move does not refund the old term at another company, and it does not refund a cyberfusionsystem term outside the 7-day window. Read the cancellation page before you assume a move is a return.