I'm not exactly an expert in quantum physics, but this seems to be a simple question, and I can't find an answer anywhere!
There are specific types of fields used in physics: scalar fields (i.e. as in the case of the Higgs boson), vector fields (i.e. as in magnetic fields), tensor fields (i.e. as in general relativity), etc. But what types of fields are used in QFT to model elementary particles? Is my confusion simply a result of me thinking in purely classical terms?
Yes, your confusion is wholly caused by you thinking classically ;)
In a hand-wavy way, particles are certain localized excitations of the quantized fields.
The QFT picture contains the particle picture in the perturbative approach known as Feynman diagrams (and, relatedly, the LSZ formalism). There, we are given the action of our theory dependent on some fields (be they scalar, fermion, vector, Rarita-Schwinger, tensor or even higher spin). An instructive model is so-called $\phi^4$-theory with the (here, massless) action
$$ S[\phi] := \frac{1}{2}\partial_\mu \phi \partial^\mu \phi - \frac{\lambda}{4!}\phi^4 $$
Particles are obtained in the asymptotic past and future ($t = \pm \infty$) by assuming that the interaction term $\frac{\lambda}{4!}\phi^4$ does not play a role when excitations of the fields are far apart, so we have a free theory there, with free action $S_0[\phi] = \frac{1}{2}\partial_\mu \phi \partial^\mu \phi$ and the classical e.o.m. allows the usual mode expansion of $\phi$ into creation and annihilation operators of particles of definite momentum, $a^\dagger(\vec p)$ resp. $a(\vec p)$. The creators/annihilators correspond to the same ladder operators in the quantum harmonic oscillator, for example, which is why one says that they represent excitations of the quantum field. Now, the Feynman diagrams/LSZ formalism tell you what happens with what probability when you let these free particles interact - they let you calculate the scattering amplitudes, which are essentially the entries in the S-matrix. The "Feynman rules" for writing down diagrams tell us that, for our $\phi^4$ action, we have as building blocks one kind of lines/particles corresponding to the scalar field $\phi$, and that only those graphs are valid which either only include these lines not crossing at all, or those that contain vertices corresponding to the interaction term $\frac{\lambda}{4!}\phi^4$, i.e. crossings of four of these scalar lines.
Now, we can also speak of virtual particles, of which the only uncontroversial thing to say is that they are internal lines in the Feynman diagrams, which do not correspond to the real particles in our free creation/annihilation spaces, but are often spoken of as particles, as well.
There are also resonances, about which I seem to recall a suitable treatment in Srednicki, but I'm not confident in proclaiming anything about them except that they are also often lumped in under the term "particles".
Fundamental fermions like quarks and leptons are described by the spinor field, while gauge bosons like photons are described by the vector field. They together with the Higgs bosons are currently what we have in the Standard Model for elementary particles.
The type of field that you have depends on the way that your field transforms. The fields that you encounter in quantum field theory usually are:
If you go to beyond the standard model stuff you'll get spin 3/2-fields and spin 2-fields (the graviton).
The QFT is strongly based on the group theory formalism. Often when people say about some QFT theory they primarily say about the symmetries of the theory - invariance of lagrangian of theory (or about covariance of equations of motion) under sets of transformations. The group theory formalize these statements and help to construct theories which corresponds to a given particles with interaction without a second thought.
So, free theories in QFT are based on irreducible (in some sense - the most elementary) representations of the Poincare group, because this group represents the locally exact (I neglect general relativity) symmetries of our space-time: isotropy and homogeneity.
Poincare group naturally includes description of mass and spin. The irreducible representations of the group is characterized by sets of numbers which are given as eigenvalues of so-called Casimir operators. In a case of Poincare group there are two Casimir operators - $P^{2}$ (square of translation operator, its eigenvalue is square of mass) and $W^{2}$ (square of Pauli-Lubanski operator, its square is square of eigen angular momentum - spin). So for quantum field $\psi$ must be true following statements: $$ \hat{W}^{2}\psi = -m^{2}s(s + 1)\psi , \quad \hat{P}^{2}\psi = m^{2}\psi , \quad \psi{'}_{ab}(x) = \hat{T}(N)_{ab}^{\ cd}\psi_{cd}(\Lambda^{-1} (x - a)). $$ Here $m, s$ are mass and spin respectively and $T(N)$ is matrix of the Poincare group transformations.
The interaction theories of these fields must satisfy requirements of lorentz-invariance and causality.
After some "manipulations" you will get the statement that field with $n $ undotted spinor indices and $m$ dotted spinor indices which is symmetrical under indices permutations represents particles with spin $s = \frac{n + m}{2}$. The necessary condition for the theory to be lorentz-invariant and causal is that half-integer spin particles have Fermi-Dirac statistics (fermions) while integer-spin particles have Bose-Einstein statistics (bosons).
In Standard model the most of elementary stable particles are fermions, while fundamental interactions are represented by bosons. The fundamental interactions are constructed by the special way - corresponding theory must be locally gauge invariant under sets of $SU(n)$ transformations. The roots of this assertion lies in the fact that free electromagnetic field $A_{\mu}$ theory is gauge invariant.
So the conclusion is following: the most elementary theories of fundamental interactions is characterized by sets of symmetries represented as direct product of Poincare group and internal symmetries group.
