Secure multi-party computation enables multiple mutually distrusting parties to perform computations on data without revealing the data itself, and has become one of the core technologies behind privacy-preserving machine learning. In this work, we present several improved privacy-preserving protocols for both linear and non-linear layers in machine learning. For linear layers, we present an extended beaver triple protocol for bilinear maps that significantly reduces communication of convolution layer. For non-linear layers, we introduce novel protocols for computing the sigmoid and softmax function. Both functions are essential building blocks for machine learning training of classification tasks. Our protocols are both more scalable and robust than prior constructions, and improves runtime performance by 3-17x. Finally, we introduce Morse-STF, an end-to-end privacy-preserving system for machine learning training that leverages all these improved protocols. Our system achieves a 1.8x speedup on logistic regression and 3.9-4.9x speedup on convolutional neural networks compared to prior state-of-the-art systems.