Partial moons are great for viewing. I almost never look at the moon, instead preferring Venus (same crescent as the moon!), Mars (ice caps!) and Saturn (over the course of my life, I've seen the ring system tilt). Not to mention the moons of Jupiter. Or some really bright nebulae.

However, for sharing photos or for people new to astronomy, the full moon is a very good target. Craters and shiny mountaintops are difficult to understand at first, but the large mare and Tycho crater are very prominent. Over the course of an hour, you can show them how the moon moves not only in relation to the Earth (because the telescope needs adjusting) but also relative to the background stars. And lastly, they can look up at the moon after they've pulled their eye away from the eyepiece, and still see detail. For that moment on, they'll look at the moon with a sense of familiarity - not just a disk in the sky but now an intricate object which they had one seen in intimate detail.