"We wait for them to play."
Translation:Noi aspettiamo che suonino.
Interesting question. In Italian (I'm native) the difference is that "Aspettiamo loro per giocare" means that when they arrive, we'll play with them. While "Aspettiamo che loro giochino" implies that we won't play, but we're going to watch them playing. So I'm curious to know how you translate it in English.
"We're waiting to play with them" is the first, "We're waiting for them to play" is (mostly) the second. One thing that distinguishes English, as an almost uninflected language, is the importance of word order. There's an old linguistics joke about that, the difference between "house cat" and "cat house."