I speak from the perspective of a CS student, planning on double-majoring in Math. I find that a lot of my core courses are fairly predictable; these tend to be part of "standard progression". In a similar way as we have Algebra 1 -> Algebra 2 -> Precalc -> Calc -> Multivar/Diffeq as the standard math progression (at least in Maryland), much of what we encounter in the first two years is setting us up for something later.
This is why I'd typically consider gen-eds to be the most surprising. Sort of like electives in high school, but there is such a variety of them that one can find some fairly specific things to take classes on. Two examples from me would be Ethics of AI and Down the Rabbit Hole; they are more discussion-centered and encourage thinking. To find such courses, I'd advise just scrolling the gen-ed pages on Testudo and see what comes up. We have to do gen-ed anyways -- might as well make them fun ones.
Excluding a lecturer from last semester who I already knew, I'd say my biggest success was with my Ethics of AI teacher. I participated a fair bit in-class, so I'd say I was fairly visible.
As a whole, it seems the hardest part about connecting with faculty is knowing where to look. Most of my early classes had a lot of students (although perhaps this is because I'm a CS major), and in settings like that, my lecturers taught hundreds of students, so they couldn't get to know their students very well. My suggestion would be to find a fairly "small" setting (could be a club, organization, or perhaps a particularly discussion-centered class) and participating in it in a way that faculty notice.
Strangely, I'd say I didn't have much of a problem adjusting to the new setting here at UMD. I'd say I didn't connect much with other students, primarily because of a lack of interest in doing so; another possible obstacle would be that the setting isn't conducive to socializing, such as during lectures. However, I grant that the interactions from many connections may lead you to opportunities and other information you wouldn't otherwise have.
Primarily knowing what to do with my time. I am fairly efficient with getting things done, but this means a lot of time left over, and I often found (and still do, to an extent) that I had a lot of time that I didn't know what to do with. My advice would be to look for extracurriculars to join; perhaps they may inspire you to pick up a new hobby, and at the minimum, participating in something is more productive than doing nothing.