I started an internship a few days ago, and I'm very happy with the learning experience I'm getting so far! I think this is something students should have in mind since the first year, and I don't recommend waiting until your last quarter. Starting doing internships for experience as early as possible will help you gain experience to build your resume, will help you develop new skills, and will help you understand how searching for a job is and prepare yourself for job interviews.
Here's a list of five ways in which you can find internships in NYC (and most of them applicable to many other places.)
This is a very popular website. It can be sketchy some times, and you should be very careful when using it because virtually anyone can post anything there. However, it is very useful. You don't have a profile here, you can just see the listings. Go to the jobs section and find whatever industry you're interested in. Once you're in that section, check the box that says "Internship" and click on search again. When you find something you're interested in, you will need to reply that post through e-mail and that's where you will attach your resume and cover letter. Check it out at newyork.craigslist.org.
In an ideal world, you would have a list of companies you would love to work for, and they would have internships available. Specially with big companies, they usually have a "careers" section on their company website that allows you to apply to their job openings. You can browse those and look for internships available. It can also be that you apply through their website and your resume gets lost somewhere in a remote server... but you never know!
Finally, I would like to mention LinkedIn as a great tool to find internships. You can search for internship positions with their search engine, but you can also contact key people in your network who can help you find one. This can also be a great way to start networking and I'm sure employers would appreciate the fact that you made the effort to find an opportunity this way. I actually have a couple friends who have succeeded doing it this way!
Remember that no one will knock on your door and tell you you're the super intern they're looking for. You need to knock on their door and tell them you're the super intern they need! Good luck on your internship hunting! Thanks for reading and see you soon!