My answer really depends on where one is traveling. In most cases, I suggest using a credit card as often as possible. Credit cards subject you to the least amount of risk (banks cut off fraudulent use very quickly), and you get the most favorable exchange rates.
I usually take two cards, in the event one is declined for some reason. One card is sufficient if you are traveling with someone who also has a card that can be used as your back up. Just make sure the cards have a large enough credit line to cover your big expenses (hotel, car rentals, etc.). Keep in mind some locations require a deposit that may eat into your credit availability.
Advise your bank or credit card company where you will be traveling; otherwise it is almost guaranteed that your card will be declined, especially after the first purchase. These days notification can be done online. Also be sure to use cards without a foreign transaction fee. Those pesky fees add up quickly if you use your card often.
Another question that comes up quite often is the need for a PIN when using a credit card. Quite frankly, I have never needed a PIN for regular purchases. However I have heard of some locations requiring it. I request a PIN from my card company just in case.
Credit cards, though, are not perfect in all situations. When traveling to more remote areas or smaller towns that don't attract as many tourists, shops (smaller shops especially) may not accept credit cards. In many towns, including big cities, payment with credit cards require a minimum purchase, so if you are wanting a few post cards or a bottle of water, you'll either have to purchase more items or use cash. Credit cards are also the least best option to get local currency as they often come with higher fees and interest charges.
So what about debit cards and cash?