Before You Leave The UK
It goes without saying that when you leave your current address in the UK, you will likely need to redirect mail. For example, most UK financial institutions are required to send you a statement every 6 months in the post. It's possible you forgot to notify some of of these institutions about your move, so...
Set up a mail redirection service, ideally for 12 months if you can afford it.
The Royal Mail can do a redirection to your new overseas address, but there are limitations. For example, stuff that's too big or heavy will not be sent overseas. Neither will they redirect Special Delivery or Recorded Delivery items.
You can set up a redirect or change an existing one from overseas, again, with limitations designed mainly to prevent ID theft and fraud/money laundering.
You can find out more from the Royal Mail at this link: Royal Mail Redirection
Suggestion: See if a family member or good friend will act as a UK destination for your mail redirection. If comfortable give them permission to open any mail. Stuff that is really urgent they can phone/email/text you about so you deal with it quickly, otherwise you could be waiting 2-4 weeks for the International redirect from Royal Mail. This also means they can bin any junk mail etc you don't need. Anything else that is of interest/importance to you but not urgent they can package up for you and send on to you - I'm sure you'll reimburse them for the expense!
If you are moving in to (very) short term accommodation on arrival in the US, before finding a more permanent residence, you may wish to hold off on redirecting your mail to the USA, and have it redirected to a UK address instead. The Royal Mail can hold your mail for you, for up to 2 months. Fees start at £8.95 (as of 30 Aug 2010) for up to 17 days holding. It can take 5 days to set up. See more at Royal Mail Holding.
 Protecting Your UK Identity
Identity theft can be a problem in the USA as well as the UK. To help avoid the possibility that while out of the UK somebody else steals your identity, register with the main credit reference agencies such as [www.equifax.co.uk Equifax], [www.experian.co.uk Experian], or CallCredit who all offer a warning service for a reasonable monthly fee. They'll email and/or text you anytime something changes on your credit file (eg address etc) or a credit search is done.
 On Arrival In The US
Once you arrive in the US, and assuming you have achieved some stability in where you will be living, there's a couple of things you can do to make things a little easier and less stressful for you.
In some states or cities it may be mandatory to let the US Postal Service or USPS know that you've moved in. Either way, it is a good idea to pop down to your local Post Office and just let them know. It's never a bad thing to be on personal terms with the people who look after and deliver your post for you! (see here for examples: examples)
Not every house or apartment has a mail slot or letterbox in the front door. Often there's a mailbox at the end of your path running from your front door to the street. In buildings with several flats or apartments, it's possible everybody's mailbox is in the same place. Sometimes these mailboxes are lockable, but the quality of locks can vary. See examples here.
Also know that in many places the Mail Man (or woman) will pick up any mail you want posted, if you leave it in your mail box. Just know that this approach isn't secure - this is no reflection on the USPS employee - simply that anybody could access your mailbox if they tried...
At some point you may find yourself getting lots of junk or spam mail - much of which can contain many of your personal details. You can opt out of this by visiting https://www.optoutprescreen.com/ - just like the Mail Preference Service in the UK. Incidentally you can do the same with spam phone calls by going to https://www.donotcall.gov/.
You can find out much more about sending and receiving mail in the US by visiting http://pe.usps.com/text/dmm100/intro.htm
 Protecting Your Identity In The US
Sign up for the warning & ID theft protection service with the main credit agencies in the US (Experian, Equifax), same as for the UK.
Avoid sending any mail containing your personal details by leaving it in your mailbox for the Mail Man/Woman to collect - go direct to the Post Office and send it from there, or use a courier service such as UPS or FedEx.
Any mail you receive that contains your details and that you no longer need, shred, using a cross-cutting shredder before you bin it.
Sign up for the anti spam mail service at https://www.optoutprescreen.com/ which will cut down if not eliminate the amount of mail going around with your personal details on it.