We present the BaLROG (Bars in Low Redshift Optical Galaxies) sample of 16 morphologically distinct barred spirals to characterize observationally the influence of bars on nearby galaxies. Each galaxy is a mosaic of several pointings observed with the integral-field unit (IFU) SAURON leading to a tenfold sharper spatial resolution (˜100 pc) compared to ongoing IFU surveys. In this paper we focus on the kinematic properties. We calculate the bar strength Q__b from classical torque analysis using 3.6-μm Spitzer (S4G) images, but also develop a new method based solely on the kinematics. A correlation between the two measurements is found and backed up by N-body simulations, verifying the measurement of Q__b. We find that bar strengths from ionized gas kinematics are ˜2.5 larger than those measured from stellar kinematics and that stronger bars have enhanced influence on inner kinematic features. We detect that stellar angular momentum `dips' at 0.2 ± 0.1 bar lengths and half of our sample exhibits an anticorrelation of h3-stellar velocity (v/σ) in these central parts. An increased flattening of the stellar σ gradient with increasing bar strength supports the notion of bar-induced orbit mixing. These measurements set important constraints on the spatial scales, namely an increasing influence in the central regions (0.1-0.5 bar lengths), revealed by kinematic signatures due to bar-driven secular evolution in present-day galaxies.