  Although instruments for measuring the radial velocities (RVs) of stars now
routinely reach sub-meter per second accuracy, the detection of low-mass
planets is still very challenging. The rotational modulation and evolution of
spots and/or faculae can induce variations in the RVs at the level of a few m/s
in Sun-like stars. To overcome this, a multi-dimensional Gaussian Process
framework has been developed to model the stellar activity signal using
spectroscopic activity indicators together with the RVs. A recently published
computationally efficient implementation of this framework, S+LEAF 2, enables
the rapid analysis of large samples of targets with sizeable data sets. In this
work, we apply this framework to HARPS observations of 268 well-observed
targets with precisely determined stellar parameters. Our long-term goal is to
quantify the effectiveness of this framework to model and mitigate activity
signals for stars of different spectral types and activity levels. In this
first paper in the series, we initially focus on the activity indicators
(S-index and Bisector Inverse Slope), and use them to a) measure rotation
periods for 49 slow rotators in our sample, b) explore the impact of these
results on the spin-down of middle-aged late F, G & K stars, and c) explore
indirectly how the spot to facular ratio varies across our sample. Our results
should provide valuable clues for planning future RV planet surveys such as the
Terra Hunting Experiment or the PLATO ground-based follow-up observations
program, and help fine-tune current stellar structure and evolution models.
