You should make your current utilities supplier aware that you are moving house to see if it is possible to move your existing tariff to the new property if you are happy with what you are already paying.
Bear in mind if the new place if more or less energy efficient your bills might vary from what you currently pay.
Take your Meter Reading on the day you leave and use your phone to take a photograph of the meter as evidence of your reading in case of disputes.
When you get to your new place you need to do the same thing. Find out who the supplier is and take a reading and photograph of the meter. This will help make sure there are no disputes between you and the previous owner.
Be wary of any supplier ‘Standard Tariffs’ which are usually the most expensive that the supplier might have you on when you first move into your new property.
If possible shop around when you move into your new place and continue to shop around regularly to make sure you are getting the best tariff.
Consider as well the age of the boiler within the property and find a registered Gas Safe engineer to visit your new home to service it to ensure it is in good and safe working order and not leaking deadly carbon monoxide.
You can find registered Gas Safe Engineers online using their search feature.