  Cluster mergers leave distinct signatures in the ICM in the form of shocks
and diffuse cluster radio sources that provide evidence for the acceleration of
relativistic particles. However, the physics of particle acceleration in the
ICM is still not fully understood. Here we present new 1-4 GHz Jansky Very
Large Array (VLA) and archival Chandra observations of the HST Frontier Fields
Cluster Abell 2744. In our new VLA images, we detect the previously known
$\sim2.1$ Mpc radio halo and $\sim1.5$ Mpc radio relic. We carry out a radio
spectral analysis from which we determine the relic's injection spectral index
to be $\alpha_{\rm{inj}} = -1.12 \pm 0.19$. This corresponds to a shock Mach
number of $\mathcal{M}$ = 2.05$^{+0.31}_{-0.19}$ under the assumption of
diffusive shock acceleration. We also find evidence for spectral steepening in
the post-shock region. We do not find evidence for a significant correlation
between the radio halo's spectral index and ICM temperature. In addition, we
observe three new polarized diffuse sources and determine two of these to be
newly discovered giant radio relics. These two relics are located in the
southeastern and northwestern outskirts of the cluster. The corresponding
integrated spectral indices measure $-1.81 \pm 0.26$ and $-0.63 \pm 0.21$ for
the SE and NW relics, respectively. From an X-ray surface brightness profile we
also detect a possible density jump of $R=1.39^{+0.34}_{-0.22}$ co-located with
the newly discovered SE relic. This density jump would correspond to a shock
front Mach number of $\mathcal{M}=1.26^{+0.25}_{-0.15}$.
