Learn Italian: Pago con una banconota, non con la carta. - I pay with a banknote, not with the card.

Yes, carta normally means paper, but in the context of shopping and payments, it is commonly used as a short way to say carta di credito (credit card) or carta di debito (debit card).

I know verbs ending in -gare like pagare sometimes add an h. Why isn't there one in pago?

You only add an h to verbs like pagare when the ending starts with an i or an e, to keep the hard G sound (for example, tu paghi). Because the letter o already creates a hard G sound, pago doesn't need an h.

In American English, a banknote is called a 'bill'. Could I use the Italian word conto here instead of banconota?

No, you have to use banconota for paper money. While conto translates to 'bill', it refers to the check you ask for at a restaurant to see how much you owe, not the physical money you use to pay it.

Why does the sentence say una banconota (a banknote) but la carta (the card)?