  We investigate the properties of galaxies as they shut off star formation
over the 4 billion years surrounding peak cosmic star formation. To do this we
categorize $\sim7000$ galaxies from $1<z<4$ into $90$ groups based on the shape
of their spectral energy distributions (SEDs) and build composite SEDs with
$R\sim 50$ resolution. These composite SEDs show a variety of spectral shapes
and also show trends in parameters such as color, mass, star formation rate,
and emission line equivalent width. Using emission line equivalent widths and
strength of the 4000\AA\ break, $D(4000)$, we categorize the composite SEDs
into five classes: extreme emission line, star-forming, transitioning,
post-starburst, and quiescent galaxies. The transitioning population of
galaxies show modest H$\alpha$ emission ($EW_{\rm REST}\sim40$\AA) compared to
more typical star-forming composite SEDs at $\log_{10}(M/M_\odot)\sim10.5$
($EW_{\rm REST}\sim80$\AA). Together with their smaller sizes (3 kpc vs. 4 kpc)
and higher S\'ersic indices (2.7 vs. 1.5), this indicates that morphological
changes initiate before the cessation of star formation. The transitional group
shows a strong increase of over one dex in number density from $z\sim3$ to
$z\sim1$, similar to the growth in the quiescent population, while
post-starburst galaxies become rarer at $z\lesssim1.5$. We calculate average
quenching timescales of 1.6 Gyr at $z\sim1.5$ and 0.9 Gyr at $z\sim2.5$ and
conclude that a fast quenching mechanism producing post-starbursts dominated
the quenching of galaxies at early times, while a slower process has become
more common since $z\sim2$.
