Your name server information is not updated during a domain transfer. To avoid problems, we strongly recommend that you update your name server information at your old registrar before you initiate a domain transfer.
If you are obtaining DNS from your old registrar and you don't update your name servers, they will probably remove your DNS service when they process your transfer. This typically manifests as your domain suddenly going to the old registrar's parking page or failing outright after the transfer completes.
To find out how to update your name servers on another registrar's system before a transfer, contact them directly.
If you have already transferred a domain without changing the name servers and you now need to update them, please see this related FAQ entry.