Pointing a telescope solely at the lit portion will produce a quick case of eye fatigue. Look for craters, mountain ranges, and the flat plains of frozen lunar lava called maria. Each evening during the next week, the moon will get a bit higher in the sky. It will also get a bit fatter. As the terminator marches across the face of the moon, new details will be exposed along its length.
Most bookstores sell astronomy books that include detailed lunar maps.
It's fun to identify for yourself the various craters and mountains that you're seeing each night. The best telescopic planets are nicely placed for observing during the early evenings of late December and early January. Jupiter is certainly the best planet to observe in telescope or binoculars. Binoculars will show three or four of its brightest moons lined up around the planet. A telescope will show the moons and a lot more.