If you're moving home
Tell your supplier you're leaving — shortly before you go, or on the day. Ask how much notice they need. If you've already moved, tell them as soon as you can, because the account can keep billing in your name until they update their records.
They'll want your move-out date, a forwarding address and a meter reading taken on the day you leave. Photograph the meter while you still have access — thirty seconds now beats arguing about an estimate later.
Without a reading, your supplier may bill you on an estimate, which can mean paying for more than you actually used.
Ask when your final bill will arrive. If you're in credit, that money often goes back to the account your Direct Debit came from. Suppliers have to take steps to return credit on a closed account, and you can ask for a refund at any time — if they refuse, ask them to explain why.
One people don't expect: leaving your old contract early can carry an exit fee. Ask your old supplier whether one applies when you move — some charge one, some waive it.
If you're staying put and moving to a new supplier
You don't normally need to contact your current supplier at all — your new one arranges the changeover and the two of them settle it between themselves.
Check your current tariff for an exit fee before you start, give your new supplier a meter reading when they ask for one, and leave your Direct Debit in place until the final bill from your old supplier has been settled.