I swear I'm an old soul or something because I've always found myself fitting in more with my teachers than with my peers. I've always liked talking to my teachers in elementary school all the way through high school and especially now in college with my professors.
I was called a teacher's pet in middle school, partially included in all the bullying regarding me skipping a grade. People assumed I had an "in" with the teachers because even though I skipped a whole year of schooling, I was still getting better grades than a lot of my classmates. Maybe that's also partially what pushed me to be friendly with my teachers, I don't know. Which brings me to another important point.
I never understood why people weren't nice to their teachers. What makes students think that teachers/professors aren't worth their respect? I was raised to treat everyone nicely, and I think about that almost every day. All it takes sometimes is a smile and I forget that some people weren't raised to give even that much.
I think I made real connections with some of my high school teachers because of the nature of our relationships. I was in the band with my robotics teacher, sang with my choir instructor and band director, was involved in plays with the drama teacher and SAT prep teacher. Apart from just trying to be a good student, I sought to make lasting relationships with some really awesome people.
Now that I'm in college, these relationships are even more important for me to have. I really enjoy talking to some of the professors here, both in my major and out of my major. I've made friends with a few of them that I purposely took a second class with because I enjoyed having them the first time.
A not-so-hidden benefit of being friends with professors is being able to take a class with them. Let me explain. Say you need to take a certain class, and there are two or three different professors teaching it. You know the class isn't necessarily going to be fun, so you opt to take it with the professor you already know. That way, you at least have a good professor teaching it and you'll likely feel more comfortable going to them for extra help if you need it.
In the business world nowadays, you can't seem to get anywhere unless you know someone. Having connections has become more important than having the actual qualifications needed for jobs. If you already know professors who are in the field you're looking to go into, you have a leg up on the next candidate for a position, perhaps even earning the opportunity to have one of those professors write a letter of recommendation for you.
Even if the positive relationship only provides you with a more mature and experienced person to talk to about your life or future career, you will still find the benefits outweigh any possible negatives. It never hurts to be on your professor's good side.