You should be able to get a free debit card from any bank where you have an account, if you're over 18 years old.
For credit cards, very few of them will actually have a monthly fee, and they tend to be either from smaller firms or cards with specific benefits. What you're probably thinking of is the APR, or Annual Percentage Rate, which is the % of what you have bought with the card that you have to pay in interest every month. You will never have to pay interest as long as you pay your credit card bill in full every month.
Think of a credit card like a loan; because that's exactly what it is. Every time you buy something with a credit card you are taking out a loan for that money, and if you can't pay it back at the end of the month, you have to pay interest.