  In this paper, we determine the relativistic and nonrelativistic energy
levels for Dirac fermions in a spinning conical G\"odel-type spacetime in
$(2+1)$-dimensions, where we work with the curved Dirac equation in polar
coordinates and we use the tetrads formalism. Solving a second-order
differential equation for the two components of the Dirac spinor, we obtain a
generalized Laguerre equation, and the relativistic energy levels of the
fermion and antifermion, where such levels are quantized in terms of the radial
and total magnetic quantum numbers $n$ and $m_j$, and explicitly depends on the
spin parameter $s$ (describes the ``spin''), spinorial parameter $u$ (describes
the two components of the spinor), curvature and rotation parameters $\alpha$
and $\beta$ (describes the conical curvature and the angular momentum of the
spinning cosmic string), and on the vorticity parameter $\Omega$ (describes the
G\"odel-type spacetime). In particular, the quantization is a direct result of
the existence of $\Omega$ (i.e., $\Omega$ acts as a kind of ``external field or
potential''). We see that for $m_j>0$, the energy levels do not depend on $s$
and $u$; however, depend on $n$, $m_j$, $\alpha$, and $\beta$. In this case,
$\alpha$ breaks the degeneracy of the energy levels and such levels can
increase infinitely in the limit $\frac{4\Omega\beta}{\alpha}\to 1$. Already
for $m_j<0$, we see that the energy levels depends on $s$, $u$ and $n$;
however, it no longer depends on $m_j$, $\alpha$ and $\beta$. In this case, it
is as if the fermion ``lives only in a flat G\"odel-type spacetime''. Besides,
we also study the low-energy or nonrelativistic limit of the system. In both
cases (relativistic and nonrelativistic), we graphically analyze the behavior
of energy levels as a function of $\Omega$, $\alpha$, and $\beta$ for three
different values of $n$ (ground state and the first two excited states).
