  A former member of the Hyades cluster, $\iota$ Horologii ($\iota$ Hor) is a
planet-hosting Sun-like star which displays the shortest coronal activity cycle
known to date ($P_{\rm cyc}\sim$ 1.6 yr). With an age of $\sim$$625$ Myr,
$\iota$ Hor is also the youngest star with a detected activity cycle. The study
of its magnetic properties holds the potential to provide fundamental
information to understand the origin of cyclic activity and stellar magnetism
in late-type stars. In this series of articles, we present the results of a
comprehensive project aimed at studying the evolving magnetic field in this
star and how this evolution influences its circumstellar environment. This
paper summarizes the first stage of this investigation, with results from a
long-term observing campaign of $\iota$ Hor using ground-based high-resolution
spectropolarimetry. The analysis includes precise measurements of the magnetic
activity and radial velocity of the star, and their multiple time-scales of
variability. In combination with values reported in the literature, we show
that the long-term chromospheric activity evolution of $\iota$ Hor follows a
beating pattern, caused by the superposition of two periodic signals of similar
amplitude at $P_{1} \simeq 1.97 \pm 0.02$ yr and $P_{2} \simeq 1.41 \pm 0.01$
yr. Additionally, using the most recent parameters for $\iota$ Hor b in
combination with our activity and radial velocity measurements, we find that
stellar activity dominates the radial velocity residuals, making the detection
of additional planets in this system challenging. Finally, we report here the
first measurements of the surface longitudinal magnetic field strength of
$\iota$ Hor, which displays varying amplitudes within $\pm4$ G and served to
estimate the rotation period of the star ($P_{\rm rot} =~7.70^{+0.18}_{-0.67}$
d).
