The sustainability of the planet and its inhabitants is in dire danger and is among the highest priorities on global agendas such as the Sustainable Development Goals (SDGs) of the United Nations (UN). Solar energy -- among other clean, renewable, and sustainable energies -- is seen as essential for environmental, social, and economic sustainability. Predicting solar energy accurately is critical to increasing reliability and stability, and reducing the risks and costs of the energy systems and markets. Researchers have come a long way in developing cutting-edge solar energy forecasting methods. However, these methods are far from optimal in terms of their accuracies, generalizability, benchmarking, and other requirements. Particularly, no single method performs well across all climates and weathers due to the large variations in meteorological data. This paper proposes SENERGY (an acronym for Sustainable Energy), a novel deep learning-based auto-selective approach and tool that, instead of generalising a specific model for all climates, predicts the best performing deep learning model for GHI forecasting in terms of forecasting error. The approach is based on carefully devised deep learning methods and feature sets through an extensive analysis of deep learning forecasting and classification methods using ten meteorological datasets from three continents. We analyse the tool in great detail through a range of metrics and methods for performance analysis, visualization, and comparison of solar forecasting methods. SENERGY outperforms existing methods in all performance metrics including Root Mean Square Error (RMSE), Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE), Forecast Skills (FS), Relative Forecasting Error, and the normalised versions of these metrics. The proposed auto-selective approach can be extended to other research problems such as wind energy forecasting and predict forecasting models based on different criteria (in addition to the minimum forecasting error used in this paper) such as the energy required or speed of model execution, different input features, different optimisations of the same models, or other user preferences.