If you stay on site consider using your room key to pay for everything in the parks. You can then use whatever means you want to pay off your balance. We use travers checks.
You can use credit cards at the sit down restaurants, shops, etc. (INCLUDING Counter Service restaurants). The only places you can NOT use a credit card are the free-standing carts (ice cream carts, drink carts, etc.). Some of those MIGHT even take a card, but just to be safe I would make sure to have a little bit of cash on me!!!
There are a few places that are stationary but don't take credit cards, but not many. The beaver tail stand is one of them. I've heard that the place where you can buy food at Fantasmic is another. But the vast majority of places that don't actually move do take cards. All of the counter service restaurants definitely do--it's just some of the stands that don't.
Actually VISA is the new official card at WDW, AMEX is no longer the official card although it is still accepted. I hear that soon they will offer a Disney VISA that accumulates points toward Disney perks! Where do I sign!?!