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Normal and tangential contacts between a cylindrical steel indenter (wheel) and an elastomer possessing high adhesive properties have been investigated. Computer simulation of the indentation and detachment processes has been performed for the case of indentation in the normal direction, which demonstrated good agreement with experimental data. For the rolling-friction regime, the adhesive component of the friction force has been determined by analyzing the measured dependences of the tangential component of the contact force on wheel displacement. A case of sliding friction with blocked wheel rotation has been considered. In the presence of adhesion, the sliding-friction force is proportional to the contact area. In the absence of adhesion (the elastomer is completely coated with chalk dust), stick-slip friction is implemented. The frequency and amplitude of stick-slip transitions depend on the indentation depth in the elastomer.