Gravity is pulling us and the ground down. But, unless something bad is happening, neither we nor the ground under us is moving because another force is pushing us and the ground up. That force is electromagnetism.
Think of your foot pushing down on the ground. You can envision a top layer of atoms pushing back up. (Real life is messier than nice flat layers of atoms, but just imagine.) That layer is being kept apart from the layer of atoms below it by the electromagnetic forces between the electrons in the atoms. That layer pushes down and is pushed up by the layer below it, and so on all the way down. In fact, it's not long before you are a tiny fraction of the weight of one layer of atoms.
Heavier things, like stars, need stronger forces to support themselves from collapsing under their own weight because electromagnetism isn't strong enough. For normal stars, the main force deep down is called radiation pressure -- which is just the force of the light shining on the star's matter, pushing it up. When the star burns out, it might use electron degeneracy pressure, which is the quantum-mechanical pressure created by electrons not wanting to be in the same place. If it's too heavy for that, it might become a neutron star, which is supported by the same type of pressure, except with neutrons. And if it's too heavy for that, it collapses into a black hole under its own weight.