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 (similar to 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-mu m Spitzer (S(4)G) 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 Qb. 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 h(3)-stellar velocity (nu/sigma) in these central parts. An increased flattening of the stellar a 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.