The Moon reaches its first-quarter phase on March 27 at 07:43 UT. At first quarter, the Moon rises in the east near mid-day, reaches its highest point as the Sun sets, and remains well positioned for viewing in the mid-evening hours with binoculars or a telescope. This time of year at first quarter, the Moon lies in the northerly reaches of the ecliptic in the constellation Gemini. So it’s as far above the horizon as it gets for northern-hemisphere observers.
As always, the best viewing on the Moon lies along the terminator, the line between night and day. Here the long shadows make it easier to see surface features such as mountains and craters. At first quarter along the terminator, the lunar Caucasus Mountains are spectacular about one-quarter the distance from the northern limb of the Moon to the southern limb. The immense lava-filled crater Plato, a little further north still, also begins to emerge over the next day or two. And further south, in the heavily cratered lunar highlands, the large crater Albategnius stands out about one third the way from the south limb to north limb.
When the Moon is full, it appears fully lit from our point of view. So it’s strange that when it’s half lit, we call it first (and last) quarter. But don’t worry too much about terminology. Go out and have a look for yourself!