Yes on credit cards. And I agree that the family membership is the way to go. Although, if you're only going once and plan on many family members going, look into group rates. You must reserve two weeks in advance for a group rate, but it could be worth it depending on your situation
Yes you can use a credit card they have birthday packages but they're not discounted and there is no discount for a whole family only if you get the membership which is the past you get for going to the zoo for her free for a whole year as many times as you want it's best that you get that for your whole family so that way you can go as many times as you want and get in free
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