An experiment to study double beta decay processes in Cd using a CdWO crystal scintillator (mass 215 g) enriched in Cd at 66 is in progress at the Laboratori Nazionali del Gran Sasso (LNGS), in Italy. Events in the CdWO detector are recorded in (anti)coincidences with two large-volume CdWO scintillation counters. The design of the detector system, calibration and background measurements, methods and results of data analysis to determine key detector characteristics are described. The experimental data are compared with Monte Carlo simulation results, and a background model is constructed. The radioactive contamination of the setup components is studied. After 3 yr of data taking the experimental sensitivity to decay processes in Cd is at the level of lim T yr.