In the framework of the GIARPS High-resolution Observations of T Tauri stars (GHOsT) project, we study the accretion properties of 37 classical T Tauri stars of the Taurus-Auriga star-forming region (SFR) with the aim of characterizing their relation with the properties of the central star, with jets and disk winds, and with the global disk structure, in synergy with complementary ALMA millimeter observations. We derive the stellar parameters, optical veiling, the accretion luminosity (L_acc_), and the mass accretion rate (dM_acc_/dt) in a homogeneous and self-consistent way using high-resolution spectra acquired at the Telescopio Nazionale Galileo with the HARPS-N and GIANO spectrographs that are flux-calibrated based on contemporaneous low-resolution spectroscopic and photometric ancillary observations. The L_acc_-L*, dM_acc_/dt-M* and dM_acc_/dt-M_disk_ relations of the Taurus sample are provided and compared with those of the coeval SFRs of Lupus and Chamaeleon I. We analyzed possible causes for the observed large spreads in the relations. We find that (i) a proper modeling in deriving the stellar properties in highly spotted stars can reduce the spread of the dM_acc_/dt-M* relation, (ii) transitional disks tend to have lower dM_acc_/dt at a given M*, (iii) stars in multiple systems have higher dM_acc_/dt at the same M_disk_, (iv) the dM_acc_/dt versus disk surface density has a smaller spread than the dM_acc_/dt-M_disk_, indicating that opacity effects might be important in the derivation of M_disk_. Finally, the luminosities of the OI 630nm narrow low-velocity component and high-velocity component (HVC) and the deprojected HVC peak velocity were found to correlate with the accretion luminosity. We discuss these correlations in the framework of the currently accepted models of jets and winds. Our results demonstrate the potential of contemporaneous optical and near- infrared high-resolution spectroscopy to simultaneously provide precise measurements of the stellar wind and accretion wind properties of young stars.