© ESO 2020.Context. RZ Cas is a short-period Algol-type system showing episodes of mass transfer and δ Sct-like oscillations of its mass-gaining primary component. This system exhibits temporal changes in orbital period, v sin i, and the oscillation pattern of the primary component. Aims. We analyse high-resolution spectra of RZ Cas that we obtained during a spectroscopic long-term monitoring lasting from 2001 to 2017. In this first part we investigate the atmospheric parameters of the stellar components and the time variation of orbital period, v sin i, and radial velocities (RVs), searching for seasonal changes that could be related to episodes of mass exchange and to a possible activity cycle of the system triggered by the magnetic cycle of the cool companion. Methods. We used spectrum synthesis to analyse the spectra of both components of RZ Cas. The study of variations of the orbital period is based on published times of primary minima. We used the least-squares deconvolved (LSD) binary program to derive separated RVs and LSD profiles of the components. From the LSD profiles of the primary we determined its v sin i. Using Markov chain Monte Carlo simulations with the PHOEBE program, we modelled the RV variations of both components. Results. Spectrum analysis resulted in precise atmospheric parameters of both components, in particular in surface abundances below solar values. We find that the variation of orbital period is semi-regular and derive different characteristic timescales for different epochs of observation. We show that the RV variations with orbital phase can be modelled when including two cool spots on the surface of the secondary component. The modelling leads to very precise masses and separation of the components. The seasonal variation of several parameters, such as v sin i, rotation-orbit synchronisation factor, strength of the spots on the cool companion, and orbital period, can be characterised by a common timescale of the order of nine years. Conclusions. We interpret the timescale of nine years as the magnetic activity cycle of the cool companion. In particular the behaviour of the dark spots on the cool companion leads us to the interpretation that this timescale is based on an 18-yr magnetic dynamo cycle. We conclude that the mass-transfer rate is controlled by the variable depth of the Wilson depression in the magnetic spot around the Lagrangian point L1. In the result, based on available data, we observe a damped activity cycle of the star, starting with a high mass-transfer episode around 2001 with a calculated mass-transfer rate of 1.510-6 Mpdbl yr-1, followed by quiet periods in 2006 and 2009, slightly higher activity around 2013 and 2014, and again followed by quiet periods in 2015 and 2016. However, owing to missing data for years 2010 and 2011, we cannot exclude that a second high mass-transfer episode occurred within this time span.