What are quarks made of? So atoms are formed from protons and neutrons, which are formed from quarks.
But where do these quarks come from? What makes them?
 A: The standard mainstream answer is to consider them as fundamentals. Another standard, but not mainstream, answer is that we call generically "preons" to the hypothetical components of quarks and leptons. The most stablished -arguably- preon theory is Harari-Shupe, sometimes referred to as "rishon theory", but there are others.
Without preons, 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. Theorists have also proposed models where the states are Rishons.
The 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 favourited theory.  
A: I cannot resist this mother goose quote:

What are little boys made of?
What are little boys made of?
Frogs and snails,
And puppy-dogs' tails;
That's what little boys are made of.
What are little girls made of?
What are little girls made of?
Sugar and spice,
And all that's nice;
That's what little girls are made of.

You state :

So atoms are formed from protons and neutrons, which are formed from quarks.
  and ask:
  But where do these quarks come from? What makes them?

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.
A: 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.
A: First, let me emphasize that nobody knows what to expect for sure when we probe at smaller and smaller distances (or at higher energies) the until now as elementary considered elementary fields: the electron ($e^-$), the electron neutrino ($\nu_e$), the up-quark ($u^{\frac {+2}{3}},$), and the down-quark($d^{\frac{-1}{3}}$), together with their second and third generation, the massive $W^{+/-}$ and the $Z^0$  (the force carriers of the weak force), the Higgs (explaining mass), and the hypothetical superheavy X- and Y-bosons, which (according to the theory) enable the proton to decay, and have an electric charge of $+\frac{4}{3}$ resp. $+\frac{1}{3}$.
A very plausible guess though is contained in Harari's Rishon theory (which already has been mentioned in a previous answer), which can account for all reactions between elementary fields (except the ones that involve the Higgs field). With only two elementary fields, the T-rishon, and the V-rishon, a preon theory can't get more economical (it's impossible to construct the until now known elementary fields out of just one field). This is certainly more elegant than the manifold of elementary particles which prevails nowadays. I name "elegance" because some physicists regard this as an argument in favor of new ideas (by the way, I don't).
Other arguments in favor of this theory:


*

*The decay of the proton is explained very easily:
$uud (=p)→d\bar{d}(= \pi^0) + e^{+}$
$ u  +  u→  \bar{d}+e^+$
$TTV+TTV→TVV+TTT$ 

*The theory states that the amount of matter is equal to the amount of anti-matter

*According to the Rishon theory, the weak force isn't a fundamental one, just like the "old" strong force that was once thought to be transmitted by the pion (to be compared with the $W^{+/-}$ or the $Z^0$) turned out to be a residual force, and the fundamental strong force is now known to be conveyed by massless gluons.


The Higgs field(s) has no place in this theory, which seems a major throwback since they may have been discovered. It is said that because of this all elementary fields will have no mass. Both rishons are massless, but when they form bound states (the only state they can be in), then maybe the force (the one conveyed by hyper color gluons) between them is thát big that they can (despite the velocity of light they travel with) stay together and form massive fields. If so, what to do with the Higgs field? Well, maybe, in that case, we can use this economic theory to disproof the existence of this Goddamned particle field. Like I wrote in the comment below:

To me, the Higgs-mechanism is a rather artificial construct and therefore I'm inclined to say that the evidence for the Higgs is contaminated. So one can use the Rishon theory to disproof the existence of the Higgs field.

A: 
What are quarks made of?

We don't know what quarks are made up of, it maybe that we've touched bottom here, or that further structure is yet to be uncovered. 
So far, the results of the LHC in uncovering further structure - apart from the discovery of the Higgs boson - has not been encouraging. It maybe that our current technological capability is simply not up to the mark. 
Recall, that classical mechanics was reinvigorated when Galileo looked through a telescope at the night sky, and the theory of black body radiation using the then thermodynamical theory of Boltzmann gave results at variance with experiment and prompted Planck's introduction of the atomic hypothesis into energy, ie the quantum hypothesis. 
It maybe we will simply have to wait for further technological ingenuity before we can properly address physics beyond the SM, and this by the looks of it, could be some wait. 
However, one of the main current contenders for explaining quarks is string theory; in fact, string theory first arose as a theory of the strong force as a kind of flux tube that connected quarks; it should be pointed out, as all the major practitioners of the theory caution, that this theory is highly speculative, as one ought to expect when we are so far away from a regime that is directly accessible to experiment.
