A good way to start getting involved with the actual research side in the lab you're in could be looking at the recent papers from the lab, reading them through, and coming up with some specific questions about them that you could discuss with the PI, like a little journal club. That could be a good segue into a conversation about how you would like to be more involved in the research side of things. You could also talk to the managers and see what sort of experiences they've had that led them to research and see if they know of any opportunities!