Credit and debit cards are the easiest way to pay in America, since bank transfers are not really an option for merchants. It's really uncommon to pay by check at the point of sale (to the extent where people might respond negatively), and cash is actually pretty inconvenient in the US due to odd-even pricing tactics giving everyone pockets full of small coins.

Most people would agree that the most intelligent way to use a credit card is to treat it as a debit card and never incur a balance. Instead, you'd pay each month's bill in full. This way, you pay no interest on your debt; in fact, you are effectively incurring zero debt. Moreover, a lot of credit cards offer rewards programs in terms of cash back or "points" as a percentage of your expenditures. With some mixing and matching of cards, you can easily get 2-3% cash back on your aggregate purchases, or rack up some free airline flights.

I actually find this to be a huge moral gray area; the reason that Visa, Discover, and MasterCard can afford to pay out handsome rewards is partly because of the financially illiterate charging more to their credit cards than they can afford to pay off. [1] I don't have the exact numbers on me, but I think interest versus merchant fees was about a 67%-33% split in terms of credit card company revenue. (American Express is an exception because they do not allow you to carry a balance on a lot of their cards; they will report your account as delinquent and terminate service.)

Otherwise, as other posters have mentioned, credit cards offer fraud protection. Cards aimed at people with good credit scores often come with other nice benefits, such as additional reimbursement if an airline loses your bag.

[1] There are valid reasons to place a huge charge that you can't afford to pay off immediately. Emergencies do happen, for instance, and it's really great to have a line of credit for that situation. But then your credit card serves the same purpose as a traditional loan.