The north east and south east quadrants are those which first come into view following New Moon when the sun once more rises on the side of our satellite that faces us.
Mare Crisium, the waterless Sea of Crises, is one of the first great lava plains to come into view, followed by the forked plains that include Mare Tranquillitatis, the Sea of Tranquility, where Man first walked on the Moon in the Apollo 11 mission. They are visible to the unaided eye but binoculars will show them clearly.
Interesting features to look for with telescopes include the brilliant Proclus near Mare Crisium’s western edge. Look in Mare Serenitatis, the Sea of Serenity, for two spots marking smaller craters, Bessel and Linne. A larger feature known as the “ghost crater” because it has been flooded with lava is Lamont in Mare Tranquillitatis.
As the lunar phase approaches First Quarter, conditions become ideal for telescope users to spot the spectacular Alpine Valley, a gash that cuts through 130km high mountains. The dramatic Apennine mountain range also begin to come into view at this time.
Update! For an alternative, inverted view, here is a chart with south at the top.
You can download a check sheet for the entire list of 50 features here! (PDF file).