First, you could talk to a professor from one of your current courses/projects. Just find them at the end of a class and explain how you're interested in this course and why you'd like to get involved; then ask if there are any opportunities in their lab or with other of their colleagues. You can also email professors from past courses: start with the course they taught so that they feel a personal connection.
In my opinion this is the best way to reach a professor because they will know you have relevant skills and you can impress them with your coursework.
But it's not always easy to find a match between a course you really like and a professor whose research you're interested in. So another way is to lookup professors from your university on the internet (you can start from the university website and browse research labs, then professors/labs usually have their own websites with a lot of information on their current work). That's what I did when I was in grad school looking for a research assistantship. I contacted a lot of professors, got a few replies, meetings and found a great RA position.
Keep in mind that these professors are usually busy and don't have a lot of time to read such emails, or may be contacted by a lot of other students. To make sure you pop out and provide useful information, make sure to:
- do your homework: read about the professor/lab's research work, papers, goals etc on their website (or search via Google scholar)
- introduce yourself properly: describe your research interests, skills (like courses you are attending) and background. And attach a resume.
- connect the dots: clearly explain why you're reaching to this professor (vs. any other one in the field) - maybe you read about a project you want to get involved in, or you found one of their papers really interesting
- be direct: tell them what you want. For example, number of hours you can work with them, if you need a paid position or if you are ok to volunteer etc..
This may seem like a lot, but believe me this will pay off. Good luck!
In my freshman year, I emailed my assigned adviser asking about any opportunities in the department. I made sure to specify that I wanted to do it as undergrad research experience and did not need to get paid. He didn't have any opportunities but he forwarded my request on to the whole department. That really made me cringe but the upside is that I got a research opportunity, and later research funding out of it. SO I say start by emailing a professor!
I recommends the following next steps: