Does Costco still accept American Express cards? If not what kind of credit cards do they accept? Do I have to apply for the Costco card?

No, since their deal with Citigroup, Costco now only accepts Visa. You do not need to have their branded Visa, but they will not accept American Express. They will accept any Visa card. The whole debacle was so they could save money on swipe fees and instead they have outraged many customers. It is still yet to be determined if this was a good move for them. The Visa rule includes debit cards, but you can pay with a check from any bank. Those who had American Express cards will automatically receive the new Visa and it will have no effect on their credit score.