Not all apps are created equal. In fact, not all apps are available to you in the US. There are other-country-specific apps, as well as apps that are "soft launching" in smaller markets -- but what if you want to download those apps?
You can change your country in the App Store, but going back and forth gets to be kind of a hassle, since you'll have to input new payment information each time. Or you can create a second Apple ID for a different country, and just sign in and out of the store. Here's how:
Step 1: Sign out of iCloud
To create a new iTunes account, you'll first need to sign out of your current iTunes/iCloud account. Do this by opening Settings and tapping iCloud. Scroll to the bottom of the screen, where you'll see a button that says Sign Out. Tap Sign Out to sign out of your iTunes/iCloud account -- your iPhone will warn you that if you sign out of your account, all photo stream photos, documents, and data stored in iCloud will be removed from your iPhone.
Your iPhone will also prompt you to choose whether to keep iCloud data and contacts on your iPhone, and then asks you to enter in your iCloud password to turn off Find My iPhone.
Step 2: Create a new account
Once you're all signed out of your iTunes/iCloud account, you can create a new account. Go to Settings > iCloud and tap Create a new Apple ID.
You'll be asked to enter in a birthdate, name, and email address (you'll need to enter in a different email address from your other iTunes/iCloud account). You will also be asked to choose a password, pick and answer three security questions, and input an optional rescue email address. Once you've filled everything out, tap continue (this may take a moment).
Step 3: Verify your account
Check your email to find Apple's verification message. Click on the link in the email message to verify your email your address. Apple will ask you to sign into your new iCloud account to verify your address.
Step 4: Choose the country on your new iTunes/iCloud account
Before you can start grabbing apps from another country, you'll need to pick the country on your new account. To do this, open the App Store and tap Featured. Scroll down to the bottom of the page, where you'll see a button that says Sign In. Tap this button and sign in using your new iTunes/iCloud credentials.
You should see a pop-up informing you that you haven't used this Apple ID on the iTunes store. Tap Review and wait a few seconds, and a list of countries will appear on the screen. Choose the country you want your new ID to be associated with, and then tap Next. Apple will ask you to agree to some terms and conditions, tap Agree.
Next, Apple will ask for your billing info. In order to move forward, you will need to input a billing address and phone number that matches the country you chose, but you do not need to add a credit card at this time. So you can use any address -- a government building, a bank, a hotel. You won't be able to download paid apps without a credit card, but this way you'll at least get to download free apps from that country.