The graduate engineer trainee's roles and responsibilities would depend on the specific company. However, overall, a trainee is put through training first to equip them with the basic skills and then they're usually accepted in ongoing projects. The typical training period is about 3-6 months. Once the training period is successfully completed, the engineers are then offered a regular designation (Junior Engineer, Technical Staff, etc.).
From there, the engineers are evaluated annually or biannually on the basis of their performance and awarded promotions & better compensations. Career growth would typically depend on your performance in the company. However, it's too early to discuss career growth. I'd recommend focusing on acquiring a variety of skills and concentrating on becoming a better engineer. That's the most important thing that helps a *lot* in the longer run.