We investigate clustering properties of dark matter halos and galaxies to search for optimal statistics and scales where possible departures from general relativity (GR) could be found. We use large N-body cosmological simulations to perform measurements based on the two-point correlation function (2PCF) in GR and in selected modified gravity (MG) structure formation scenarios. As a test-bed, we employ two popular beyond-GR models: $f(R)$ gravity and the normal branch of the Dvali-Gabadadze-Porrati (nDGP) braneworld. We study a range of simulated halo and galaxy populations and reveal a noticeable MG signal in the monopole and quadrupole moments of the redshift-space 2PCF, and in the so-called clustering wedges. However, once expressed in terms of the linear distortion parameter, $\beta$, the statistical significance of these signals largely diminishes due to a strong degeneracy between MG-enhanced clustering and modified tracer bias. To circumvent this, we consider statistics less dependent on the bias: relative clustering ratios. We generalize the monopole ratio proposed in earlier work to multipole moments and clustering wedges, and introduce a new estimator of the $\beta$ parameter. The clustering ratios we extract foster noticeable differences between MG and GR models, reaching a maximum deviation of 10\% at 2$\sigma$ significance for specific variants of $f(R)$ and nDGP. We show that such departures could be measured for $\beta$ if non-linear effects at intermediate scales are correctly modeled. Our study indicates that the clustering ratios give great promise to search for signatures of MG in the large-scale structure. We also find that the selection of an optimal tracer sample depends on a particular statistics and gravity model to be considered.