Question

When you jump, you push down on the earth and it pushes back up against you. The earth pushing up against you is what causes you to go into the air. Why doesn’t your push cause the earth to go down if your push on the earth is equal and opposite of the earth's push on you?

Answer 1

That's a great question !

The answer is: It does !

A push on an object causes the object to accelerate in the direction of the force.  

The less mass the object has, the more the force accelerates it.

Now, when you jump, the forces on you and on the Earth are equal forces.

The up force on you causes you to accelerate up by some amount.

The down force on the Earth causes the Earth to accelerate down by some amount.

The Earth's mass is something like 5,972,000,000,000,000,000,000,000 kg, while your mass is something like 50 kg.

The Earth has something like 119,400,000,000,000,000,000,000 times as much mass as you have.

So your acceleration is something like 119,400,000,000,000,000,000,000 times as great as the Earth's acceleration.

==> The Earth's downward acceleration, caused by your jump, is there.  It's just too small to notice.

BUT . . . That's the reason why seismometers (instruments to detect and measure the vibrations from distant earthquakes) have to be located as far as possible from cities and busy roads.

In places that are too close to cities and roads, the Earth's surface is always vibrating, wiggling, jiggling, heaving and weaving, in reaction to the forces of people walking around, cars and trucks driving around, even rain falling down.  And kids jumping up and down !  

In such places, these people-motions are louder and stronger than the vibrations coming from distant earthquakes.  Seismometers wouldn't work there.