Have extra keys made, if necessary. Make an appointment for handing over and collecting the keys, for your old home and your new home respectively.
Pack things you don't need very often and take them to your new home already.
Fridge and freezer
They must be defrosted a couple of days beforehand.
Children and pets
Arrange for them to be looked after and house your pets temporarily with family or friends: it will be one thing less to worry about on the day.
Safety deposit box
You can temporarily store precious possessions such as money and jewelry in a safety deposit box in the bank. This will make you feel a lot safer.
Take out your painting tools and start redecorating your new home, putting furniture in place, etc.
You must of course also do some cleaning, both in your old and your new home. This way you'll leave your old home nice and tidy while getting your new home in good shape as well.