You can't feel it, but the Earth is moving around the Sun at 66,000 miles per hour. One trip around, or one year, is approximately 365.2422 days; just a bit less than 365.25. This motion has visible and measurable effects.
First - as the Earth moves, the Sun appears to move around the sky. This means that different constellations are visible at different times of year. You see Orion in the winter and Scorpius in the summer. Here's a diagram of the effect.
There's another effect which concerns the day. What is a day? It is one rotation of Earth with respect to something. There are two possible somethings.
Since the apparent position of the Sun against the background stars depends on the Earth's location, it is the Earth's orbit that determines the Sun's path among the stars. This is the Ecliptic. You can see it on our large sky globe. You will notice that the Ecliptic does NOT coincide with the Celestial Equator; it is inclined to the equator by 23.5 degrees. The Sun is South of the Equator for half the year and North of it for the other half.
The Zodiac is the classical band of 12 (modern 13) constellations that the Sun passes through during the year. These constellations do not correspond to the sun signs of astrology (more on that shortly).
If you think that it's hot here in summer because the Earth is closer to the Sun in summer consider this: Earth is at perihelion (closest to the Sun) about January 4. It is farthest from the Sun in early July. There's got to be something else going on. If distance from the Sun was the primary cause, we should be hot in January and cold in July. What's the deal? (This diagram shows the details)
If you thought that the apparent size of the Sun should change with Earth's varying distance from it, you'd be correct. This composite picture of the Sun shows the slight apparent size change.
Remember that 23.5 degree inclination of the Ecliptic with respect to the Celestial Equator? That's where the secret lies. This is the result of the orientation of Earth's axis, which is NOT perpendicular to the orbit plane. The axis is tilted 23.5 degrees away from perpendicular. This is the situation in Northern hemisphere summer. This diagram has some more detail.
This tilt produces several effects.
Longer term: Precession is the slow wobble of the Earth's axis. It makes our North Celestial Pole (and Celestial Equator) move. The period of the precessional wobble is about 26,000 years, which is why you never noticed it. We happen to have a very nice pole star (Polaris) right now. A mere 2,000 years ago there was no accurate pole star, although 4,800 years there was a very good one. Here's a good diagram showing the path of the North Celestial Pole.
There's another effect. Since our geocentric celestial coordinates (declination and right ascension) are based on Earth's axis, the wobble causes the coordinate system to wobble, too. This means that the positions of stars in that system change. Astronomers make new star charts about every 50 years. We are now using star charts for epoch 2000. Thirty years ago the charts were epoch 1950. Astronomers are the only ones who have to worry about this; it does affect high-precision telescope pointing.
Remember that the zero point for right ascension is the intersection of the Ecliptic and Celestial Equator at the March 21 position of the Sun. As the Equator (and indeed the entire coordinate system) wobble with precession, this zero point slides westward along the Ecliptic. This has resulted in sun signs (related to dates) sliding westward one constellation in about 2,000 years. This effect, discovered in ancient times, posed a problem for astrology. If your sign is Aries, the Sun is actually in Pisces on your birthday.
The Moon is the ONLY thing that orbits around the Earth - the only natural object, that is. There's a LOT of stuff in orbit that we put there. We're not counting all that stuff.
It has long been known that the Moon goes around the Earth. Its motion is nearly uniform and its apparent size doesn't change (much). The sidereal period of the Moon is roughly 27.3 days, and its synodic period is about 29.5 days. The synodic period is the time between full moons.
The Moon changes its appearance in a regular cycle. We used a little model to illustrate the day and night sides of the Moon. One side of the Moon (and any other object as well) is dark because the Moon (object) itself is in the way. The night side of the Moon is NOT in Earth's shadow.
A demonstration with a lamp and a plastic sphere illustrated the Moon's phases and their cause.
A lunar eclipse really does involve the Earth's shadow. A solar eclipse involves the Moon's shadow. The section on eclipses in the book is quite good. Remember that if the Moon's shadow actually reaches Earth, a total eclipse of the Sun occurs. If the shadow does not reach all the way to Earth, the eclipse is annular.
We don't get eclipses every month because the Moon's orbit does not lie in the ecliptic plane. At new moon, the Moon is usually above (north) or below (south) the Sun. At full moon, the Moon passes north or south of Earth's shadow. Only at times 6 months apart, when the new and full moon positions are IN the ecliptic plane, can eclipses occur.
Speaking of eclipses, we used a demonstration, which shows how shadows behave when the light source is bigger than the object making the shadow. There is a region of total shadow where sunlight is completely blocked, and a region of partial shadow where the sunlight is only partially blocked. The total shadow is called the umbra while the partial shadow is the penumbra. See Figure E-16. You get a total lunar eclipse when the Moon moves into Earth's umbra. This doesn't happen at every eclipse. Sometimes the Moon is far enough off center in the shadow that only part of the Moon gets into the umbra; this makes a partial eclipse.
During a lunar eclipse the Moon usually (not always) remains easily visible, glowing with a noticeable reddish color. This might be puzzling - if the sunlight is blocked by Earth, what is lighting the Moon? Nothing magic - it's still the Sun. Earth has an atmosphere, which refracts some sunlight and bends it into the shadow. The blue light is blocked by the atmosphere, leaving the red light to go on to the Moon. If you were standing on the Moon, you would see a solar eclipse, with the Earth blocking the Sun. There would be one major with respect to our solar eclipses: there will be a very bright glowing ring around the Earth. That's the light source. If Earth had no atmosphere (like the Moon), the Moon would go completely dark during eclipse. Of course, there wouldn't be anyone here to watch!