If you're moving home
Tell your supplier you're leaving, shortly before you go or on the day, and ask how much notice they need. If you've already moved, tell them as soon as you can, because the account stays 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, so photograph the meter while you still have access. Without a reading, your supplier may bill you on an estimate. Ask when your final bill will arrive and, if you're in credit, how it will be refunded. Leaving a fixed deal early can carry an exit fee, so ask your old supplier whether one applies when you move.
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.