Speech emotion recognition (SER) plays a significant role in human-machine interaction. Emotion recognition from speech and its precise classification is a challenging task because a machine is unable to understand its context. For an accurate emotion classification, emotionally relevant features must be extracted from the speech data. Traditionally, handcrafted features were used for emotional classification from speech signals; however, they are not efficient enough to accurately depict the emotional states of the speaker. In this study, the benefits of a deep convolutional neural network (DCNN) for SER are explored. For this purpose, a pretrained network is used to extract features from state-of-the-art speech emotional datasets. Subsequently, a correlation-based feature selection technique is applied to the extracted features to select the most appropriate and discriminative features for SER. For the classification of emotions, we utilize support vector machines, random forests, the k-nearest neighbors algorithm, and neural network classifiers. Experiments are performed for speaker-dependent and speaker-independent SER using four publicly available datasets: the Berlin Dataset of Emotional Speech (Emo-DB), Surrey Audio Visual Expressed Emotion (SAVEE), Interactive Emotional Dyadic Motion Capture (IEMOCAP), and the Ryerson Audio Visual Dataset of Emotional Speech and Song (RAVDESS). Our proposed method achieves an accuracy of 95.10% for Emo-DB, 82.10% for SAVEE, 83.80% for IEMOCAP, and 81.30% for RAVDESS, for speaker-dependent SER experiments. Moreover, our method yields the best results for speaker-independent SER with existing handcrafted features-based SER approaches.