Michael is offering great advice.
Being on the job and hungry is important and you’ll find that the designers around you will gladly share their skills and experience if you are showing willingness to learn.
What I would also like to add is to find a balance.
Let the prospect of a fulltime hiring motivate you but not paralyze you. It can feel daunting. It certainly did for me.
The important part is to just push through that and to show that you fit well with the team.
For me and a bunch of other interns at the agency that hired me after an internship, it was the fact I “won” a project.
In other words, my concept was selected by the design director and the client.
If that happened during your internship, it usually meant you were being kept on.