Accurate surface roughness prediction is critical for ensuring high product quality, especially in sectors such as manufacturing, aerospace, and medical devices, where the smallest imperfections can compromise performance or safety. However, this is very challenging due to complex, non-linear interactions among variables, which is further exacerbated when working with limited and imbalanced datasets. Existing methods leveraging traditional machine learning algorithms require extensive domain knowledge for feature engineering and substantial human intervention for model selection. To address these issues, we propose a Neural Architecture Search (NAS)-Driven Multi-Stage Learning Framework, named NASPrecision. This innovative approach autonomously identifies the most suitable features and models for various surface roughness prediction tasks and significantly enhances the performance by multi-stage learning. Our framework operates in three stages: (1) architecture search stage, employing NAS to automatically identify the most effective model architecture; (2) initial training stage, where we train the neural network for initial predictions; (3) refinement stage, where a subsequent model is appended to refine and capture subtle variations overlooked by the initial training stage. In light of limited and imbalanced datasets, we adopt a generative data augmentation technique to balance and generate new data by learning the underlying data distribution. We perform extensive experiments on three distinct real-world datasets, each associated with a different machining technique, and compare with various machine learning algorithms. The experimental results underscore the superiority of our framework, which achieves an average improvement of 25.7%, 22.7%, and 39.1% in terms of Mean Absolute Error (MAE), Root Mean Square Error (RMSE), and Standard Deviation of Residuals (STDR), respectively. This significant performance enhancement not only confirms the robustness of our framework but also establishes it as a generic solution for accurate surface roughness prediction. The success of this approach can lead to improved production efficiency and product quality in critical industries while also reducing the need for extensive domain knowledge and human intervention.