If they don't take Visa did you verify that they will take the other types of cards? When my kids were in college I could only pay the tuition by credit card if I wanted to pay an additional fee.

Presuming they will accept another card you first need to determine what you are trying to accomplish with points/mileage. Is this for flights? Hotels? If flights, where are you based, where do you hope to travel to and which airline do you feel works best for you. Many of the cards are linked to one major alliance.