How to calculate when the moon is highest in the sky to the earthbound observer?

In winter, the full moon is opposite the sun, and as the sun is low, the full moon is high.

In summer the full moon is low (for the same reasons). The crescent moon is high in summer (and low in winter) but as the crescent moon is near the sun, it is normally not visible during the day.

During spring and autumn, the sun, and the moon follow roughly equal paths, with no phase of the moon being higher in the sky.

Third quarter, being at right angles to the sun will be at an intermediate altitude, in both summer and winter.

For exact calculations either use technology, or a set of astronomical tables and a sharp pencil!