We'll publish individual guides on how to accomplish these steps in the future, but for now here is a to-do list when you move from one place to another in Japan.
Before you go
Cancel/change the address of all your utilities/credit cards etc.
Obvious, but there you are. Note that some things, like telephone or internet, are better changed than cacelled, unless you want to incur the startup fees (up around 30,000yen) again. If you're on NTT, there is even an English number you can call to do it all for you.
Get a change of address certificate from city hall (市役所 shiyakusho)
This is called 転出届け (tenshutsutodoke). Go to the foreign person desk and hopefully they speak English, if not "uhhh... hikkoshimasu" will start things moving. If you have registered your inkan stamp, you'll need to hand in the inkan proof thing they gave you.
Hand in your health insurance card (保険証, hoken shou) to your employer, and have them issue a certificate of leaving their employment (if you're not going straight into another job)
Tell the post office you're moving and give them your new address. Again "hikkoshimasu" will get the ball rolling and probably result in a change of address form to fill out.
Tie up loose ends with your landlord and make sure you thoroughly understand what money you lose from your original deposit. Note that cleaning fees cannot legally be taken from your deposit despite what your contract may say. Also, you're not needed to pay for "daily wear-and-tear" (自然消耗, shizen shoumou), so make sure you argue if they try.
At your new place
First things first, take photos of all the damage in your new accommodation and fill out the paperwork the estate agent will give you about this. Make sure he/she has copies of all the photos. If you don't do this (and, in fact, probably even if you do) be prepared to lose most or all of your deposit when you move out.
Take the change of address form and your alien card to city hall. They'll know what to do.
Apply for health insurance (at city hall) if you don't have a job, otherwise your employer should deal with this.
Change your driving licence address, and register your new parking space if necessary.
Change your car number plate with the District Transport Bureau, or 地方運輸局 (chihouunyukyoku).
Get some copies made of your keys, unpack your stuff and invite everyone to come visit. Job done.