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Conductance, on-site, and intersite charge fluctuations and spin correlations in the system of two side-coupled quantum dots are calculated using Wilson’s numerical renormalization group (NRG) technique. We also show the spectral density calculated using the density-matrix NRG, which for some parameter ranges remedies inconsistencies of the conventional approach. By changing the gate voltage and the interdot tunneling rate, the system can be tuned to a nonconducting spin-singlet state, the usual Kondo regime with an odd number of electrons occupying the dots, the two-stage Kondo regime with two electrons, or a valence-fluctuating state associated with a Fano resonance. Analytical expressions for the width of the Kondo regime and the Kondo temperature are given. We also study the effect of unequal gate voltages and the stability of the two-stage Kondo effect with respect to such perturbations.