Learn Polish: Gdzie idą goście? - Where are the guests going?

The singular is gość (guest). Since a group of guests usually includes at least one male, it falls under the virile (masculine personal) plural category. Masculine personal nouns that end in a soft sound like -ć usually form their plural by adding -e.

Why do we use idą instead of idzie?

idą is the they form of the verb iść (to go). Because goście is plural (they), we must use the matching plural verb instead of the singular he/she/it form (idzie).

If the guests were taking a taxi, would we still use idą?

No. The verb iść (and its forms like idą) specifically means to go on foot. If the guests were traveling by any type of vehicle, you would use the verb for riding or driving, saying Gdzie jadą goście?

Can I change the word order and say Gdzie goście idą?

Yes, you can, because Polish word order is very flexible. However, placing the verb immediately after the question word (Gdzie idą...) is the standard and most natural-sounding way to ask this question.