Can I buy apple care not at the apple store? (Target, etc?)

As long as it is within 30 days of the purchase, yes.

Depends. In many countries now, you cannot buy regular AppleCare for an iPhone. You can buy AppleCare+ for one, but you can only get AppleCare+ through Apple.

In the USA see http://store.apple.com/us/product/S4575LL/A/applecare-for-iphone as in the USA, there no longer is AppleCare for iPhones, only AppleCare+ which is a different product (AppleCare+ combines extended warranty with Accidental Damage coverage).

You can get AppleCare+ by phone or at an Apple Store, but must do so at the time of purchase or within 30 days of purchase. If buying at an Apple Store, they will inspect the phone for pre-existing damage, and if over the phone, you must submit it to a remote diagnostic test.

I just bought AC+ over the phone for a day old 5S. It was quick and painless.

Hi Lawrence, I was also thinking about buying AppleCare+ over the phone. Did they just have you pay by the credit card that's on your Apple ID, or is it more painful than that?

It was a separate credit card transaction, not charged to my Apple ID. Just like any telephone/web order transaction.