Learn Spanish (Latin America): Yo pido el café y pago la cuenta.

Pedir is a stem-changing verb. This means that the main vowel in the word (the stem) changes when you conjugate it. In the present tense, the e in pedir changes to an i for most forms, including the "I" form. So, instead of saying "pedo", it correctly becomes pido.

Could I use the verb ordenar instead of pedir to say "I order"?

You might hear ordenar used for ordering food in some parts of Latin America due to the influence of English. However, pedir (which literally means "to ask for") is the most natural, traditional, and widely understood way to order food or drinks in Spanish.

Do I need to say yo again before pago?

No, you don't! Just like in English ("I order the coffee and pay the bill"), you can use yo once at the beginning, and the "I" is understood for both actions. Plus, the -o ending on both pido and pago already makes it perfectly clear who is doing the action.

Why is "the bill" translated as la cuenta? Isn't a bill el billete?

El billete is a false friend here! It is used for paper money (like a dollar bill) or a ticket (like a bus or train ticket). When you are at a restaurant or café and you want to pay for what you ate or drank, the check or bill is always la cuenta.