Two different things called moving
Moving a domain and moving a site are separate operations. A transfer changes which registrar sponsors the record at the registry. Nothing about the nameservers, the DNS records or the hosting changes unless you change them yourself.
Do them one at a time and neither will be confusing.
The checklist
- Unlock the domain with the current registrar.
- Request the authorisation code, sometimes called an EPP code.
- Make sure the administrative email in WHOIS is a mailbox you can read today.
- Start the transfer with the new registrar and pay the fee.
- Approve the confirmation email.
What the fee buys
A transfer includes a full year of renewal, added on top of the time already left on the name. A domain with seven months remaining comes out of the transfer with nineteen.
The two blockers
A domain registered or updated in the last sixty days cannot move, by ICANN rule. And an expired domain has to be renewed before it can be transferred, which is a good reason not to leave a move until the last week.