I started in a lab with my own independent project first semester of freshman year and co-authored a publication junior year. Honestly, it was 20% professionalism and 80% luck my freshman year. This may also depend on the size of your school--if you attend a larger institution, then I would say start by asking to volunteer in different labs just to get your foot in the door to learn a couple of lab techniques. If you go to a smaller school (as was my case), you should have no trouble finding a PI who you can begin an independent research project with. Don't be afraid to contact multiple professors and follow up with them if you don't hear back. You won't be misunderstood as pestering if you keep it professional (i.e. " Hi Professor X, I reached out to you a couple months ago regarding volunteering in your lab. I read your new paper on XYZ and wanted to reiterate my interest in learning 123 and eventually working on ABC. I've attached my resume for reference. I would love to meet to further discuss my interest." If you have any suggestions for future research, definitely share it! Professors love creative and innovative critical thinkers. If you still don't have any luck over the school year, then try again for summer research. Usually, that's a good time to get a lot more research done and hopefully secure fellowships for your research.
With regards to publishing, that depends largely on the lab you choose, when you join, and of course your hard work. Some labs rarely publish, other publish regularly and even support undergraduate first-authorship in smaller journals. When you join also matters. If you join at a time where the lab has accumulated a lot of publishable results, then the process may be expedited. Of course, you may not be privy to this information beforehand. However, the last part is something you can control--working hard. I've heard of PIs who were hesitant to accept undergraduate pre-med students because of poor work ethics they noticed in previous students (i.e. student never coming in to lab or not coming in regularly because they're studying for the MCAT). Don't be that person--always be grateful to be doing research at an undergraduate level, work hard, be accountable, go into lab when you say you will, regularly communicate and interpret your results with your PI, take a stab at proposing next steps, and don't be afraid to ask for opportunities to join senior lab members at an upcoming conference or other opportunity to present your work. These aspects not only lead to developing a good relationship with your PI, but will also be useful as you begin developing your career.