Even though you are physically present in the US, Amazon and Apple may be suspicious when you try to switch to the US App Store or use Amazon MP3 with your existing account. If you have no or little history with Amazon or iTunes, starting a new account with a Amazon or iTunes gift card is easy enough, but you will encounter others services that want proof of your location.
You are going to run into the situation where you need to provide a credit card with a US billing address - possibly even a card number from a US bank. You will generally get a credit card with or soon after your bank account, but that might take a while. In the mean-time, you would like to enjoy the online services you have coveted for so long…
An easy way to do this is go to a Safeway or similar store and buy a prepaid debit card or gift card. Some cards allow you to reload them with additional funds, but you should watch out for charges on these. You will generally need to register these on a web site and update your details (name, address, and so forth).
Add this credit card to your Amazon or iTunes account. In Amazon’s case, you may need to make this the default credit card and/or default credit card for 1-Click purchases, and you may need to log out and back in again.
You probably do not want to use your prepaid or gift card for your actual purchases (either because of per-purchase charges, or because it will not have much money on it). Instead, you can buy Amazon gift cards online for yourself with your foreign credit card, or buy Amazon and iTunes gift vouchers in many stores.. This should tide you over until you get a US credit card.
(I do not recommend trying this if you are not staying in the US for some time — Apple and Amazon probably have systems to prevent abuse, and you could get your account suspended and yourself potentially into other trouble.)