“book a table” / “We'll take it!” / “do you take credit cards?”
Book can be a verb meaning to reserve: book a table, make a reservation, and reserve a table are common alternatives. Take can mean choose or accept: We'll take it! accepts the table, while Do you take credit cards? asks whether cards are accepted for payment.
“let's split the bill four ways”
Split the bill means to divide the cost, and four ways means into four equal parts. Tina jokes about dividing the payment four ways, counting their two appetites as people along with herself and Frank.
“My appetite does not have its own card, Tina.”
An appetite is the desire to eat, and its own card means a card belonging to it. When Tina wants their appetites to pay, Frank plays along with the joke that treats appetites like people: his appetite does not have a card of its own.