Okay, so no one likes a brown-noser. In fact, most people do not even like being brown-nosers. It is important, though, that you actually try to form relationships with some of your professors, even if doing so will make you seem or feel like a suck-up. There are lots of different reasons that you need these relationships, and there are a few acceptable, non-suck-up ways of going about forming them. Here's a bit of advice.
One of the main reasons that you need to form relationships with your professors is that you're going to need their recommendations someday. Whether you're applying for a scholarship that will allow you to stay at school in two years when money is tight, applying for your first job, or putting in graduate school applications, you'll probably rely on your professors for letters of recommendation. Some professors will write letters for students they barely know, but most of them prefer to get to know you a little first. Plus, a letter from a professor who actually knows your personal qualities and academic capabilities can be invaluable, since it will be a more glowing recommendation than any that could be written based solely on your performance in a single class.
Another reason to form relationships with professors is that they can hook you up with opportunities that you would otherwise be unable to touch. From fellowships to research opportunities to semesters abroad, professors are in the know about what's available for students, and a professor who likes you and wants to help you will point these opportunities out to you.
Of course, there is a wrong way and a right way to go about building a relationship with a professor. Some wrong ideas would be buying gifts for a professor you barely know, which will just leave behind a stale sense of obligation or annoyance, or staying after class every day to make small talk, which can take up a lot of time from a busy person .
Instead, you should form relationships – particularly with professors in your area of study – by taking more than one class with the same professor and by engaging in conversations with that professor not only about your class but also about other aspects of your field of study. Some professors will want to keep things strictly business and will only be mildly interested in your personal life, and others will want to know more about you as a person. When it comes to this, you just have to make your own decisions about what to bring up in conversation.
If you find that you and a particular professor have a lot in common or work very well together, consider setting up a more formal mentoring-style relationship. If you're in a research-oriented area, you may be able to get a position as a research assistant, and you may also be able to be a teaching assistant as an upper-level student. Alternatively, you could set up an independent study course with a particular professor to pursue course work that would otherwise be unavailable.
Like any other relationship, there will be give and take in your relationship with a professor. You may end up as friends, but until you start getting signals that a professor wants to get to know you on a personal level, it's best to remain professional and respectful. That said, do not worry about being too formal. With today's rules, many professors who are in the business teach because they love to get to know students, and a formal attitude might be off-putting.