Is the universe made up of particles or fields? QFT describes the electron as an excitation of the electron field. The spin of electrons create magnetic fields. So which came first? How can a particle created from a field then create its own field? Or are some fields not fundamental?
 A: 
Is the universe made up of particles or fields?

No one knows. Physics describes how the Universe behaves like, not what it actually is. To our best knowledge, the Universe behaves as if it were made of fields, not particles. This is particularly proeminent when doing QFT in a curved spacetime or in non-inertial frames of reference, which are situations in which phenomena such as the Unruh effect and the Hawking effect make it clear that, within quantum field theory, the notion of particles is observer-dependent. This is not an issue, since QFT is the quantum theory of fields, not of particles.

The spin of electrons create magnetic fields.

As I mentioned in the comments, this is incorrect. The electromagnetic field possesses degrees of freedom of its own and it is not fully determined by the sources. While we often say things such as "the electric field due to the charge $q$" or "the field created by the current $\mathbf{J}$" and etc, this is imprecise language. The sources merely interact with the electromagnetic field. While this is easily excused in static problems due to the fact that there is uniqueness of solutions upon requirement of vanishing fields at infinity, this does not generalize to Electrodynamics. For example, notice that a vanishing field and a generic electromagnetic wave are both perfectly valid solutions for the electromagnetic field in the absence of sources. Hence, the electromagnetic field is there regardless of the sources. A better statement for your phrase would be that the spin of the electron interacts with magnetic fields.
(Notice that I'm being sloppy on whether physical entities exist or not, since that is actually a problem in Philosophy)

So which came first?

Within the framework of quantum field theory, which is the best description we have of the Universe so far, the fields come first. The Unruh and Hawking effects are paradigmatic examples.

How can a particle created from a field then create its own field?

The particle does not create a field. In your spin example, you are actually describing the interaction between the electron field and the electromagnetic field. In certain limits, it is sufficiently precise to drop the quantum field theoretic details of the electron field and treat the system in terms of a particle interacting with an electromagnetic field. This is usually done in non-relativistic quantum mechanics. However, it is not a fundamental description of what is going on. Furthermore, notice that the quanta (i.e., the particle-like entities) associated with the electromagnetic field are not electrons, but photons.

Or are some fields not fundamental?

Not every field is fundamental, although, to the best of present-day knowledge, both the electron field and the electromagnetic field are fundamental. The proton field, for example, is not fundamental, in the sense protons are composite, being made of quarks and gluons.
At last, notice that while the concept of particle is not fundamental in QFT, it is often useful and makes calculations considerably easier. Similarly, even in classical theory the concept of a point-particle is troublesome (for example, one can't solve Maxwell's equations for a moving point-like charge while simultaneously considering the electromagnetic forces on the charge), but most definitely extremely useful.
If it interests you, I particularly suggest checking Sections 1.3 and 1.4 of R. Wald's Advanced Classical Electromagnetism, which are available for free in the publisher's preview. They discuss in general lines how one shouldn't view electromagnetic fields as being created by charged matter and how point-like sources are problematic even in Classical Electromagnetism. Sec. 3.3 of Wald's Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics presents a nice discussion of the particle interpretation of QFT and the later chapters of the book discuss in detail the Unruh and Hawking effects I mentioned earlier.
A: 
QFT describes the electron as an excitation of the electron field.

Quantum field theory fields are completely different than the fields of Newtonian gravity and classical electrodynamics. It is a theoretical model that is very successful in fitting existing data and observation of elementary particles, the electron in your example, and is predictive of new data and observations at the microscopic world of elementary particles and neuclei..
The electric and magnetic and classical electromagnetic fields are part of the theory for macroscopic dimensions and fit the data for these large dimensions.

the spin of electrons create magnetic fields.

It can be shown mathematically  that the classical theories emerge from  the basic quantum mechanical theories. So the mathematics of the spin of the electrons  leads at large dimensions  to classical magnetic fields.

so which came first? how can a particle created from a field then create its own field?

It is in the mathematical model of QFT  that an electron is created by an operator acting on a qft field. A model. It is the measurements and observations that are fitted by the model that leads to describing electrons as created by the operator on the field. The electrons exist in nature whether one models them or not.

or are some fields not fundamental?

In the hierarchy of models, first come the QFT fields, from which emerge the classical fields, so in that sense, if we call the QFT fields fundamental then the classical are derived. BUT this is the present theoretical model. It could be that in the future new theories would have the QFT model emerging from a more fundamental theory.
