In this paper, we propose a refined multi-stage multi-task training strategy to improve the performance of online attention-based encoder-decoder (AED) models. A three-stage training based on three levels of architectural granularity namely, character encoder, byte pair encoding (BPE) based encoder, and attention decoder, is proposed. Also, multi-task learning based on two-levels of linguistic granularity namely, character and BPE, is used. We explore different pre-training strategies for the encoders including transfer learning from a bidirectional encoder. Our encoder-decoder models with online attention show 35% and 10% relative improvement over their baselines for smaller and bigger models, respectively. Our models achieve a word error rate (WER) of 5.04% and 4.48% on the Librispeech test-clean data for the smaller and bigger models respectively after fusion with long short-term memory (LSTM) based external language model (LM).