There are two equal and opposite pairs of forces in your question. You can identify them by swapping words.
gravity: "The earth pulls you" (swap words) "you pull the earth"
contact force: "the ground pushes you", "you push the ground"
Then each object can remain stationary as follows
You: the gravity of the earth is pulling you one way, but is balanced by the upward contact force from the ground. Note, this is not an equal and opposite pair in the sense of Newton's 3rd law, those pairs are labelled 1) and 2) above - but the forces are equal and means there is no resultant accelerating force on you.
The earth: the earth is attracted towards you by gravity but also pushed the other way by the contact force, again the forces cancel, but they don't count as a Newton's 3rd law pair.
In a Newton's 3rd law pair, the two forces act on different objects.
The normal force is different to gravity, it is caused by repulsive forces, between atoms, near the surface of each object, if we try and force them too close together.