A good place to start is to arrange with all the applicable utility companies for transfer of service if moving locally, give them a disconnect date at your current home and arrange for utility service at your new location. Contact the proper companies to take care of electricity, gas, water, phones, cable, garbage service and yard waste. If you are moving away and have refundable deposits make sure to give the utility companies your new address.
Cancel or change any home services like newspaper delivery you may have.
Notify the post office of your move and fill out a change of address card. Ask the people moving in behind you to forward any mail that slips through to your old address. Send your new address to companies where you have charge accounts, subscriptions, mail order clubs, alumni organizations or school groups, associations you belong to and any past employer that needs to send you tax records. After you finish the business end of changing your address don't forget to let your friends and relatives know the new address.
*Save a phone book to take with you in case you need to tie up any loose ends.
If you are moving to a different area then you need to arrange for the transfer or obtain any medical and school records. You may also want to ask around for professional referrals for new doctors and dentist. If you have prescriptions on file at a drug store have those transferred as well. When getting medical records don't forget about your pets, you need their records also.
Run around town and pick up any dry cleaning, photos being developed, anything at a repair shop. Pick up or return anything that is borrowed.