In this paper, a new feature selection algorithm, called SFE (Simple, Fast, and Efficient), is proposed for high-dimensional datasets. The SFE algorithm performs its search process using a search agent and two operators: non-selection and selection. It comprises two phases: exploration and exploitation. In the exploration phase, the non-selection operator performs a global search in the entire problem search space for the irrelevant, redundant, trivial, and noisy features, and changes the status of the features from selected mode to non-selected mode. In the exploitation phase, the selection operator searches the problem search space for the features with a high impact on the classification results, and changes the status of the features from non-selected mode to selected mode. The proposed SFE is successful in feature selection from high-dimensional datasets. However, after reducing the dimensionality of a dataset, its performance cannot be increased significantly. In these situations, an evolutionary computational method could be used to find a more efficient subset of features in the new and reduced search space. To overcome this issue, this paper proposes a hybrid algorithm, SFE-PSO (particle swarm optimization) to find an optimal feature subset. The efficiency and effectiveness of the SFE and the SFE-PSO for feature selection are compared on 40 high-dimensional datasets. Their performances were compared with six recently proposed feature selection algorithms. The results obtained indicate that the two proposed algorithms significantly outperform the other algorithms, and can be used as efficient and effective algorithms in selecting features from high-dimensional datasets.