With the surge in the number of hyperparameters and training times of modern machine learning models, hyperparameter tuning is becoming increasingly expensive. However, after assessing 40 tuning methods systematically, we find that each faces certain limitations. In particular, methods that speed up tuning via knowledge transfer typically require the final performance of hyperparameters and do not focus on low-fidelity information. As we demonstrate empirically, this common practice is suboptimal and can incur an unnecessary use of resources. It is more cost-efficient to instead leverage low-fidelity tuning observations to measure inter-task similarity and transfer knowledge from existing to new tasks accordingly. However, performing multi-fidelity tuning comes with its own challenges in the transfer setting: the noise in additional observations and the need for performance forecasting. Therefore, we propose and conduct a thorough analysis of a multi-task multi-fidelity Bayesian optimization framework, which leads to the best instantiation--amortized auto-tuning (AT2). We further present an offline-computed 27-task hyperparameter recommendation (HyperRec) database to serve the community. Extensive experiments on HyperRec and other real-world databases illustrate the effectiveness of our AT2 method.