In my experience at two very large state flagship schools — and my sons currently at a campus of 14,000 undergrads — the notion that a large school equates with large classes and a difficulty establishing relationships with professors is largely false.
My undergrad college had well over 40,000 students. My first year, I had perhaps two classes with more than 60 students: Ecology and Evolutionary Biology and Chemistry. My other classes varied between 20 and 40 students. My honors freshman writing course had only about 15 students. And once I finished EEB and Chem, and started taking more upper-level and honors courses, my class sizes were generally between 15 and 20 students, with the notable exception of accounting, where I suffered along with around 80 kindred spirits. (And I would have suffered had there been three students, a pony, and a comfort dog.)
As for establishing connections with favorite professors, it’s just not hard. Attend class. Offer answers and ask questions. Go to office hours to delve further into the material, ask questions, get recommendations for additional reading and like courses in the department, talk about internship possibilities and job prospects in the field, or share how much you enjoyed those movies / books / TV shows the professor keeps referencing. Take advantage of university “take a professor to lunch program.” If there isn’t one, get together with another student or two and ask the instructor to coffee or lunch. That’s really all it takes to get the ball rolling, assuming you have complementary interests and/or personalities. At my school of 40,000, I had three main mentors and met with them weekly. Did an award-winning honors thesis with one, worked part-time for another, and did research for the third. And had coffees, dinners, hiking trips, movie nights (with other students), etc. One of these professors encouraged me to attend UVA Law, and my sons are now UVA undergrads. And their experience has been similar — fhey’ve both found mentors and advisors and have variously been invited to office hours chats, bird-watching on Grounds, and to various speakers’ series and other university programs
Also, one of the real advantages of a large university is the greater selection of courses offered. I had far, far more courses offered in my majors as my undergraduate school of 40,000+ simply had far more professors specializing in far more areas of study. The course offerings at my sons’ school of 14,000 seem pretty narrow in comparison.
But it is early to be worried about all of this, honestly. You’ll have enough stress to deal with next fall. Give yourself until September to worry about strategic college application planning.