I'm a science major whose taken a fair share of humanities electives, and there is a very different culture in those two broad fields. Humanities classes (specifically English, History, Government) are very easy in general, but you can get bad TF's that are arbitrary and you can't figure out how to please. As hard as that is for a perfectionist like the typical Harvard student to stomach, that's just something you have to take in stride; everybody gets some hits like that. The good part is that in general you never have to do much work in those classes- the typical Harvard student can procrastinate on papers and still do well. In economics, there is a wide variety between the joke classes and the very hard classes that are like science classes. Science classes are an entirely different world once you get past the intro level. Harvard science students are cut-throat competitive- unless you are a genius or already learned the material before coming, don't expect to get above average, and hence higher than a B, unless you do every practice problem in every book and do every old exam you can get your hands on- and do that before every quiz and every test. Once you learn how to study like that, you can succeed well. But expect to have a miserable life in the process. (There are easy science classes, but those are about as rare as hard humanities ones, and often very gratifying for one's GPA but not for one's learning experience).
If you're not careful, classes at Harvard can be extremely intense and competitive. The pre-med and pre-investment banking courses (economics, statistics, etc) can be cutthroat, so be prepared to invest a lot of time in these courses. Take a balanced courseload and don't discount the importance of humanities courses in making up a well-rounded tutorial. As an economics and pre-med concentrator, I've sometimes given the humanities courses short shrift, much to my intellectual development's dismay. I think that education at Harvard is geared toward learning, usually, but it's very much up to the student what they want to make of their college education. You can go through Harvard never having read Socrates or heard of Mahler, but Harvard's requirements make it harder to do so. The most unique class I've taken is my Economics tutorial - which consists of five students and an economics graduate student as the teacher. Meeting biweekly, classes consist of discussion of 3-4 economics papers that cover a variety of topics, from health to psychology to history to the philosophy of science. The major project is writing, basically, an economics paper, and it's been one of the most difficult things I've had to do academically thus far.
I found professors to be much more approachable than I'd been warned they'd be. The biggest obstruction to close student-professor relationships, I think, is students' feelings of inferiority and insecurity. In many classes, and even smaller sections, students are afraid to ask any questions in class that aren't ubersophisticated and perfectly worded, and a sense of relief is almost palpable whenever somebody does ask a basic question. I did form some close relationships with professors, but it took a while to get comfortable enough to feel like I could be myself and that I didn't have to be impressive and well-poised all the time. Students are generally so extracurricularly involved that they really don't study all the time, but they somehow manage to do great work anyway. Many are somewhat closeted perfectionists who actually do manage to get great grades, run homeless shelters, have romantic partners, and run 3 miles a day.
Academics can get pretty aggressive at Harvard. Being premed, those courses are extremely competitive. People are very smart and some are capable of studying very efficiently. If you don't get it right, you lag behind and it's hard to catch up because other people are getting smarter and better. I know maybe one professor somewhat well but only because I took a freshman seminar and two semester courses with him, in addition that I will soon be a course assistant for one of his courses. I feel like if you make it happen, you will get a professors attention but you have to work for it. Class participation is very much integral in sections but not so much in lectures unless they are very small. The math department, my department, has very smart people and it can get quite intimidating. You have to find your niche though and work hard to understand things, even if you feel you've come from a strong calculus background.
My professors mostly know my name. The best course I took was Social Thought in Modern America (Hist 1661) with James Kloppenberg. He gives a very challenging, comprehensive reading list and you have to read it all. But in return, he gives excellent lectures, hires quality teaching fellows and requires that you come to office hours to meet with him. Taking the course triggered a ton of intellectual conversations with my friends over meals. Our study group met every day for at least two hours for a week to prepare for the exam - it was awesome in its intensity. Students don't really talk about grades. That's probably a good thing for our health. Education at Harvard is geared towards academic life, which makes a lot of people not care since they are focused on getting a job anyway.
Being a freshman, I find that intro classes suck, especially Life Sci. Generally, other classes are good and professors are willing to be available. I find the students study at the last minute a lot because they are used to doing that in high school. Intellectual conversations tend to happen more outside of the class than inside of the class because many students have diverse interests that are not completely covered by the classes that they take. I find that there are a lot of people at Harvard who just want a job, but there are also more people here than at other schools who want to learn for its own sake. I've actually asked my friends at other schools about this and I feel as though I have more intellectual conversations here that they do at their schools.
No, my professors don't know my name--but my section leaders do, which works for me :) I think also your class size and relationship with professors depends on what type of classes you are taking (your concentration) and how involved you make yourself in the class. I personally participate in class a lot, but student participation varies. Yes, harvard students have intellectual conversations outside of class, but that's not ALL we talk about :) Sadly, I think much of the students' personal investment in their own education is motivated by the desire for a job, not learning for its own sake There are billions of cool classes!! and some really boring ones you have to take--the CORE system kind of sucks.
Harvard classes range in size from 4 to 1000+ students. The amount of direct interaction with course-leading professors varies inversely with the size of the class. Class participation is generally welcomed by the teaching staff and is thus quite common. Students also have incredibly stimulating intellectual conversations outside the confines of the classroom. Caveat: there is really no such thing as an "easy class" at Harvard. Doing well academically will require a persistent, sustained effort; indeed, my favorite classes thus far have been among my most work-intensive and challenging.
Academics at Harvard are incredibly rigorous in that you're challenged to a much higher degree than you were in high school. At the same time, there are so many different classes and concentrations ("Majors") so you can form your own path in academics. One thing to be aware of is that Harvard is a liberal arts institution and NOT pre-professional. Therefore, while you'll have groups and resources for pre-law, pre-med, finance, accounting, and consulting, you'll never find that here in official academic courses.
The professors are very nice and helpful; even the most famous of professors take the time to hold office hours. There are unlimited resources available to help students understand the material taught. Large classes are often split up into smaller sections to help students get an even better learning experience.