New developers are often under probation for the first few months and are scared that they will be fired. This fear of messing up holds them back from really showing their competence. Overcome this fear of making a mistake so you can start taking some risks.
Of course, listen and take note of the requirements, systems and procedures that everyone works by in the organization, but often the difference between an experienced developer and a green one is the absence of fear. One of the reasons contractors are so successful is that they expect to be gone within a few months, so they are not afraid to get in there and take risks. Read the detail of what you are supposed to do. Study hard in your own time to shine in the areas you don't know yet so you can surprise them by suggesting improvements. Bring something to the table.
Exception: this rule doesn't mean you go off on your own without guidance. As a graduate you probably know very little so you will be making lots of mistakes. Make sure you have a mentor and ask for regular reviews of what you are working on.