Does the Sun and Moon basically appear to "move on the sky" identically regardless of location and date?
The Moon's one month orbit makes the Moon rise about 50 minutes later each day for people who aren't at extreme latitudes. (I'll examine extreme latitudes later). The Moon's orbit is inclined by 5° with respect to the ecliptic. This means that the new Moon more or less follows the path of the Sun that very day, while the full Moon more or less follows the path the Sun took six months earlier. This is why the winter full Moon rises so high in the sky in temperate latitudes.
What about more extreme latitudes? The Amundsen-Scott South Pole Station is the epitome of extreme latitudes. At the South Pole, the Sun sets six months after it rises while the Moon sets two weeks after it rises. In winter, the moon rises about a week before full Moon and sets about a week after full Moon. In summer, the moon rises about a week before new Moon and sets about a week after new Moon.