Getting your records moved.
Before the move.
It is the ward clerk's job to get your records in the right ward. Before moving, contact your ward clerk (or current ward) to let them know your new address and when you will be moving.
After the move.
Introduce yourself to your bishop your first Sunday there. Let him know that you recently moved in and would like to meet the ward clerk. Introduce yourself to the ward clerk. He can then request your records. He may ask for your name and birthday.
This is the normal process.
What if my records are not transferring?
Step 1 is to make sure you are attending the right ward. Once that is clear, it's possible that some of these steps may have fallen through the cracks. Your best bet is to speak to the ward clerk of your new ward. Even bringing him your name, birth date and membership number on a paper may help see this through. If he has requested your records, sending over an email to your old ward bishop will be helpful in ensuring the records are transefered over in a timely manner.