Don't be nervous. The therapist/psychologist will likely take the reins with the practical stuff- introducing you to the group or allowing you to say a few words about who you are and why you are there. In the groups I've observed (inpatient and IOP), the psychologist briefed the group that I would be coming in advance, to make sure everyone was comfortable with the idea. Almost always everyone has been receptive and really quite curious about who I am and what I am studying/researching.
For the most part you won't be participating in any of the exercises. Things such as talking about weekly goals and gains/setbacks, you obviously will not be able to answer, and will not be expected to. Things like mindfulness exercises...I don't see why you couldn't participate as well- I usually did. Overall, just remember why you are there- to learn about the dynamics of group therapy and gain exposure to a (potential) clinical group of interest. Focus on what the therapist is saying and trying to achieve when he/she speaks, and focus on the patients and their problems when they speak.
I would probably only talk when you are directly asked a question, unless the psychologist tells you otherwise. It's likely not a good idea to ask follow-up questions to patients or to offer your own advice- that is simply not why you are there. Questions you have are things that can hopefully be discussed with the psychologist/therapist after the session.
Oh yeah- note taking. I always did it and no one batted an eye, but I can see how in some settings it may be problematic. I would check with your supervisor, and, obviously, just don't write down personal information.
