The 31 brightest galaxies (m_B ≤ 15 mag) inside the virial radius of the Fornax cluster were observed from the centres to the outskirts with the Multi Unit Spectroscopic Explorer on the Very Large Telescope. These observations provide detailed high-resolution maps of the line-of-sight kinematics, line strengths of the stars, ionised gas reaching 2–3 R_e for 21 early-type galaxies, and 1–2 R_e for 10 late-type galaxies. The majority of the galaxies are regular rotators, with eight hosting a kinematically distinct core. Only two galaxies are slow rotators. The mean age, total metallicity, and [Mg/Fe] abundance ratio in the bright central region inside 0.5 R_e and in the galaxy outskirts are presented. Extended emission-line gas is detected in 13 galaxies, most of them are late-type objects with wide-spread star formation. The measured structural properties are analysed in relation to the galaxies’ position in the projected phase space of the cluster. This shows that the Fornax cluster appears to consist of three main groups of galaxies inside the virial radius: the old core; a clump of galaxies, which is aligned with the local large-scale structure and was accreted soon after the formation of the core; and a group of galaxies that fell in more recently.