  The magnetic topological materials have attracted significant attention due
to their potential realization of variety of novel quantum phenomena.
EuIn$_2$As$_2$ has recently been theoretically recognized as a long awaited
intrinsic antiferromagnetic bulk axion insulator. However, the experimental
study on transport properties arising from the topological states in this
material is scarce. In this paper, we perform the detailed magnetoresistance
(MR) and Hall measurements to study the magnetotransport properties of this
material. We find that the transport is strongly influenced by the spin
configuration of the Eu moments from the concomitant change in the field
dependence of the MR and that of the magnetization below the N\'eel
temperature. Most importantly, an anomalous Hall effect (AHE) and a large
topological Hall effect (THE) are observed. We suggest that the AHE is
originated from a nonvanishing net Berry curvature due to the helical spin
structure and that the THE is attributed to the formation of a noncoplanar spin
texture with a finite scalar spin chirality induced by the external magnetic
field in EuIn$_2$As$_2$. Our studies provide a platform to understand the
influence of the interplay between the topology of electronic bands and the
field-induced magnetic structure on magnetoelectric transport properties. In
addition, our observations give a hint to realize axion insulator states and
high-order topological insulator states through manipulating the magnetic state
of EuIn$_2$As$_2$.
