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In today's production machine learning (ML) systems, models are continuously trained, improved, and deployed. ML design and training are becoming a continuous workflow of various tasks that have dynamic resource demands. Serverless computing is an emerging cloud paradigm that provides transparent resource management and scaling for users and has the potential to revolutionize the routine of ML design and training. However, hosting modern ML workflows on existing serverless platforms has non-trivial challenges due to their intrinsic design limitations such as stateless nature, limited communication support across function instances, and limited function execution duration. These limitations result in a lack of an overarching view and adaptation mechanism for training dynamics and an amplification of existing problems in ML workflows. To address the above challenges, we propose SMLT, an automated, scalable, and adaptive serverless framework to enable efficient and user-centric ML design and training. SMLT employs an automated and adaptive scheduling mechanism to dynamically optimize the deployment and resource scaling for ML tasks during training. SMLT further enables user-centric ML workflow execution by supporting user-specified training deadlines and budget limits. In addition, by providing an end-to-end design, SMLT solves the intrinsic problems in serverless platforms such as the communication overhead, limited function execution duration, need for repeated initialization, and also provides explicit fault tolerance for ML training. SMLT is open-sourced and compatible with all major ML frameworks. Our experimental evaluation with large, sophisticated modern ML models demonstrate that SMLT outperforms the state-of-the-art VM based systems and existing serverless ML training frameworks in both training speed (up to 8X) and monetary cost (up to 3X)