Believe it or not, you are just on time thinking about research. I hope the following general comment would help you some;
While there are general ways to go about gaining research experience, there are unique routes to research experience specific to each department. I am going to shed light on the former, hoping that will illuminate on some possibilities.
You can write down a list of topics you are interested in doing research on. Start from the subdisciplines first, and then based on your current knowledge of each subdiscipline, write down the topics and areas of your interest. You don't have to come up with an award winning idea at this point in time. I always suggest my students to see where they can find some comfort to their deep personal pain. This helps you out most people persevere and move forward on that specific topic or area. For example, I chose suicide prevention research. It's not uplifting or inspirational for many but got me each little discovery or finding is like a gift from heavens and that keeps me going. There more you go in depth into a you, the more and deeper unique ideas for innovation and discovery you will have. That said, there is nothing wrong with choosing a different area every year or so. That might give one a multidisciplinary outlook, which is another route to creativity. After you have your list in front of you, go look at the faculty profiles of your department and try to match your areas with each faculty at least. Then score each area/faculty entry based on relevance or fit. Then write an introductory really brief email to each faculty and just like your question here, tell them you are seeking research experience seriously and in one or two sentence, tell them you are interested in x topic and wonder if they have any observer/volunteer position in their lab. Try to also meet them even if they say no at first. When you meet them, try to see if they might refer you to their colleagues in other departments, or else, they can give you a bunch of names wherever you might want to be in the next couple of years or so. After you go over each item on your list with associated faculty members, a lot of things will have happened. In case, you can't get anything at that level, do not lose your heart. There is always a good place for the curious mind. We have all been there at some point in our history. Just remember, if you don't ask, you won't get It. Please let me know how I can be more helpful. If there site allows, you can email me privately or set up a phone call. I would be really glad to help. Good luck!