Connected vehicle technology provides promising opportunities to improve road safety, enhance traffic efficiency, and reduce fuel consumption and emissions. It has been suggested that if drivers comply with suggested recommendations, connected vehicle technology can introduce huge benefits. However, whether drivers will accept suggestions and what factors will influence their likelihood of accepting the suggestions in a connected environment have not been studied. In addition, few models have been developed to predict drivers' reactions under such conditions. This paper aims to fill the research gap by examining and modeling drivers' acceptance and behavior when receiving energy- and safety-related speed recommendations through vehicle-to-infrastructure communications. A mixed-subject-design experiment was conducted in a closed-loop test track, Mcity, with seven intersection maneuver scenarios. A generally high compliance rate to the recommended speed strategies was observed that during 72% of the events, drivers changed their intersection-approaching behavior to follow the recommendations. Mixed models were conducted to explore the impacting factors while Principal Component Analysis was used to classify subjective (i.e., self-reported) data into four categories. To predict drivers' reactions when offered a speed suggestion, Random Forests were built with 13 independent variables, derived from four categories: vehicle kinematic features, device information, driver characteristics, and subjective data. Using this model, drivers' reactions during each intersection maneuver could be predicted with a reasonably high accuracy about 87.4 m away from the intersection, where the vehicle started to receive signal phase and timing information. Findings in this study can contribute to the optimization of energy-saving algorithms and the improvement of driving safety by using connected vehicle technologies.