British Gas keep turning my power off, and it's almost impossible to contact them.

Gerry1

You need to notify BG today and tell them your new address, then read all the meters on the day your tenancy ends.  Same for water.

Note the closing reading(s) on the old meter as well as the start reading(s) on the new meter (see the yellow card fixed next to the new meter).

On the day your new tenancy starts, read all the meters yourself; don't let anyone do it for you (except an official meter reader).  Register with the existing supplier(s) and give your bank details.  You can then switch to your preferred energy supplier(s).

Gerry1

If your new property has prepayment meters, always set up your own account.  Never use a key or card given to you by a landlord or previous tenant: you could end up paying off any debts they have.

poppellerant

Don't just read the meters before you leave, take photos of the readings too and make sure the readings and serial numbers are clear.

It wouldn't be a bad idea to also take photos of the meters in your new property on the same day as you move in.

Finally submit your readings online to the your old suppler and the new supplier.  If they dispute the readings, you can show them your evidence which will support you.