The high training cost has become only affordable to big tech companies, meanwhile also causing increasing concerns about the environmental impact. This paper presents CoMERA, a **Co**mputing- and **M**emory-**E**fficient training method via **R**ank-**A**daptive tensor optimization. CoMERA achieves end-to-end rank-adaptive tensor-compressed training via a multi-objective optimization formulation, and improves the training to provide both a high compression ratio and excellent accuracy in the training process. Our optimized numerical computation (e.g., optimized tensorized embedding and tensor-vector contractions) and GPU implementation eliminate part of the run-time overhead in the tensorized training on GPU. This leads to, for the first time, 2-3\times speedup per training epoch compared with standard training.