Learn Portuguese (Portugal): A viagem de ida e volta custa vinte euros.

Most Portuguese words that end in -agem are feminine. This is a very reliable rule that applies to words like a viagem (the trip), a mensagem (the message), and a coragem (the courage).

How exactly does de ida e volta mean "round trip"?

Literally, it translates to "of going and return". The word ida is a noun meaning "departure" or "a going" (coming from the verb ir, to go), and volta is a noun meaning "return" (coming from the verb voltar, to return).

How would I say "one-way trip" based on this sentence?

You just drop the return part! A one-way trip is uma viagem de ida. If you are buying a train or bus ticket, you can ask for um bilhete de ida (a one-way ticket) instead of um bilhete de ida e volta.

If I were talking about multiple round trips, would I say as viagens de idas e voltas?

No, the descriptive phrase de ida e volta stays exactly the same. You only make the main noun plural, so it becomes as viagens de ida e volta (the round trips).