You can fill in a Royal Mail Redirection form available from the post office or on the Royal Mail's website (up to 8 people on the website) to have your mail redirected to your new address - this costs £21 and you can have it directed for each surname in the house for 3, 6, or 12 monthes (this is available in the UK and abroad). During this time, you should also contact every one you are likely to receive mail off to inform them of your new address so that they can start sending it there themselves, and so by the end of the 12 monthes you should receive all mail at your new address, and it is also best to stay at your old property a little while after moving as to ensure none of your post goes there still, as sometimes it does. I hope this helps.
Although filling out your change of address form is going to help you out, you really need to contact businesses that you are dealing with, your bank, any lenders, and any companies that you receive drop shipments from. There is a lot of manual labor, so to speak, that goes into changing your address. You just can't fill out your form at the post office and hope everyone will find you.
The best thing to do is make a list of all of the people that mail you on a regular basis, create one form letter telling them about the change of address, and sending it out all at once. This way, you are able to tell everyone that you have moved, and it will only take you maybe a couple hours to make sure that it is done.
If you live in the US you can fill out a change of address form at the post office(maybe even online). But you can also notify companies, bank, magazine subscriptions, etc so they can send your stuff to your new address.
Everyone who sends you a bill: phone, cable, satellite, city water, your landlord, bank, credit cards, magazines, clubs, AAA, your employer