So atoms are formed from protons and neutrons, which are formed from quarks.
But where do these quarks come from? What makes them?
I cannot resist this mother goose quote:
You state :
How do we know atoms are formed from protons and neutrons? We have deep inelastic scatterings which showed that the atoms have a hard core, so they are not a uniformly distributed matter. Then we have the periodic table of elements which organizes itself well counting protons and neutrons.
How do we know that protons and neutrons are formed from quarks? We have the results from painstaking experiments that showed us once more that deep inelastic scattering shows a hard core inside the protons and neutrons. The study of the interaction products organized the particles and resonances into what is now called the standard model, a grouping in families that have a one to one correspondence with the hypothesis that the hadrons (protons neutrons resonances) are composed out of quarks.
But not only. They also have gluons which hold the quarks together due to the strong interaction, and the gluons have been seen experimentally , again with scattering experiments.
This is where we are now. The LHC is scattering protons on protons, i.e. quarks on quarks at much higher energies then ever before, and we are waiting for results. The theoretical interpretation called the Standard Model, so successful at lower energies presupposes that the quarks are elementary. Due to the gluon exchanges it is hard to see how a hard core might appear in quark quark scattering to take the onion one level lower, i.e. tell us that the quarks have a core.
Even in neutrino quark scattering the gluons will interfere, if the SM theory is correct at high energies. At the moment there is no experimental indication that the quarks are not elementary.
Nature though has surprised us before, and might do it again, once high energy lepton quark scattering experiments are designed and carried out in the future. Feynman I think had said: "to see what a watch is made of one does not throw one watch on another watch and count the gears flying off. One takes a screw driver". Leptons with their weak interactions are the equivalent of the screw driver.
The standard mainstream answer is to consider them as fundamentals. Another standard, but not mainstream, answer is that we call genericalli "preons" to the hypothetical components of quarks and leptons. The most stablished -arguably- preon theory is Harari-Shupe, sometimes referred as "rishon theory", but there are others.
String theory could be also an answer but not in the line of your question; quarks and leptons would be equivalent to some string states, so not "made of", but "same as". Similarly, in Kaluza Klein theory: the quarks and leptons are expected to be special states of the compactified theory. Of course, again, this is the mainstream. Theoretists have also proposed models where the states are Rishons.
Middle way, you could have the theories that propose to produce quarks and leptons out of geometry. These theories usually worry a lot about gravity.
Last, you have the non-standard theories. I myself have one of them, the sBootstrap, and no doubt that some other people will intend to answer you by proposing their favorited theory.
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Quarks are probably not made of anything more fundamental. The idea that everything has to be made of something else is not true. Light is not made of anything else, neither is gravity. That atoms had internal stuff going on was obvious, because they are electrically neutral, and yet scatter light at definite magic frequencies. Neutrons and protons betrayed their non-elementary structure because of their magnetic moments and too-strong scattering at short distances. It is usually obvious when a particle is composite.
The quarks, on the other hand, along with the electrons, light, gravity, and the gluons and W and Z bosons, are perfectly elementary, in the sense that their interactions are described well by a renormalizable quantum field theory. If they are not elementary, it is probably at a scale where they are revealed to be a string theory excitation, a quantum black hole.
Models of composite standard model fermions were interesting because they could explain the phenomenon of generations, the repeating standard model families. But string theory gives a much more natural explanation of generations, in terms of the geometry of the compactification. There is no real motivation for substructure, even though people speculate.