Suggestions for official PhD/department visit (not interview)?
First, you are probably new at this so freaking out is pretty natural. If you'd done it ten or so times, it would feel pretty natural. Actually, a bit of physical exercise can help you calm your mind. Coffee, not so much. 

But, more seriously, I think you need to be pretty natural. You will be a colleague and people will want to be able to work with you and so, to be comfortable with you. It needn't be all technical either. "Anyone here play handball?"

But you also need to demonstrate that you are suited for the environment. People will look to you for ideas, of course, so you need to be able to answer questions both about yourself and about the field/research that is going on. 

But, it is also wise not to come on too strong in the beginning. People have their ways of doing things. If you are "too helpful" you may be hurting yourself as in "Who is this guy?". So save any suggestions you might have for "improvement" in the lab for a later time. 

I once made the mistake of giving too much advice in a new employment situation and it was resented, as I learned later. Whether it was the right advice or not wasn't important. 

Probably the best thing you can do is express a lot of interest in what is going on. But each meeting will be a bit different. If "selling" is needed it will be more likely useful with the supervisor. Be positive in all things. "How do you see yourself fitting in with these people?" 

You might also be prepared for surprises. Can you give an impromptu talk on your prior research?