I'm planning to see a professor today, in our department, he is the director of one of the labs there and a PI of a couple of projects. I knew that, he's looking for some student to join his project/s. I contacted him, he told that he's a couple of funding projects and make an appointment for me to see him. So, I'm thinking what are the best question that I may ask during this first meeting.
What is your management style? What do you contribute to the research group? How do you get funding? How do you help your students succeed? Can I meet your current students? Where are your alumni now?
Ask the prof to tell you about his work. Then ask what expertise/skill set is most important to him among his lab team. Listen closely and then look for opportunities to describe how your skill set fits what he needs in his lab.
Ask: What kind of questions is your research trying to answer? Is that a question that the field itself has asked, industry, or personal interest? My experience is in X and interests are X Y and Z. Might I be able to assist you? Where do you think an answer might take you? Is there the possibility of sharing authorship?
You have to asked these relative to what experience, interests, and ideas you might have. If the research is coming from industry, the approach to join would be quite different, than one from academic inquiry itself, for example.
In any event, my suggestions are not really a template for you to follow, but to prompt you to have a conversation trajectory in mind.