Why do the Sun, the Moon and the planets all follow the same path across the sky?

The Sun, the Moon and the planets follow the same path because the Solar System is flat. Most of its major bodies orbit the Sun in nearly the same plane, which is thought to be because the planets formed from a flat disk of dust and gas. We live inside that plane, so when we look along it we see everything strung out along one line. That line is the ecliptic, the Sun's apparent path across the sky over a year.

The planets stay close to it because their orbital planes are very close to Earth's. The Moon's orbit is inclined only 5.1 degrees to it. In fact the ecliptic has a name that points straight at the Moon: the ancients noticed that eclipses only occur when the Moon is crossing it. The Sun moves a little less than one degree eastward along the ecliptic each day, and goes all the way round in a year.

The band of sky around the ecliptic is called the zodiac. It extends approximately 8 degrees north and south of the ecliptic, and within it appear the Moon and the brightest planets. Babylonian astronomers divided it into 12 equal signs of 30 degrees, around the end of the fifth century BC, like 12 schematic months. The name refers to animals, because so many of the twelve are animals.

So if you want to find Mars, Jupiter, Venus or Saturn, you do not scan the whole sky. You trace the ecliptic, the invisible road the Sun walked that day, and you look for bright, steady lights along it.