Like when is the new moon highest in the sky? Or third quarter? How do I know this for any general moon phase? Thanks.
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!
The Moon moves round the ecliptic. At full Moon it is opposite the Sun. Hence when the Sun is furthest South (in the Northern Winter), that full Moon is that which will be furthest North in that year and so on. But plainly, the Moon moves round the ecliptic every 28 days, so every 28 days or so it is furthest North etc.
In other words there is no answer based on phase or time of year alone.