I have a doctor in the family who attended a SHYMP. That person was at least in their late 40s before they finally paid off their med school debt. Taking on big debt, especially more than needed, is extremely inadvisable.
It sounds as though you will need a full ride or close to it, if your family’s income is around $40k. If there are other significant assets, that can decrease how much financial aid you will receive from schools.
You mentioned a liberal arts college because of wanting to be able to explore various interests. As a first generation college student, you would be eligible to compete for the Grissom Scholars Program at Centre College which covers full tuition, a $5k enrichment stipend, and leaves you eligible for need-based financial aid to help cover room and board.
Centre College is part of the Colleges That Change Lives association, a group of smaller schools that provide solid academics but often have less competitive admissions. You can read more about Centre on this page (though it does need to be updated, but the body of the information remains the same).
At a school like Centre, if it felt like a good fit, you might have an easier time standing out, have a bunch of resources at the ready for you, and you’d be able to explore your interests. Obviously, there is no guarantee that you would get one of these scholarships, but I’d say your odds of getting one are at least as good as you getting into an Ivy, if not better.
I would make sure you have at least one CUNY that you could commute to and apply to some additional SUNY schools. Several have been mentioned, and I’d consider adding New Paltz to that list.
Skidmore (NY) is a school where you might want to run the Net Price Calculator. It tends to be more financially generous than some other schools, based on net prices by incomes on College Navigator, and it might have enough cachet to satisfy your family while still being a more attainable (though not necessarily a likely) option for you.
Another liberal arts college that has pretty reasonable net prices is St. Lawrence in northern New York. They are not need-blind, but I do think it would be a likely admit for you. Make sure to run the NPC.
I know you haven’t asked for help with college suggestions, and your comment about liberal arts schools may have just been an off-the-cuff remark that you didn’t really mean.
I just want to reassure you that you ARE a good student and there are good colleges out there that will want you, and have a chance of being affordable (again, depending on what the NPCs say and what your family indicates they will contribute). The best way I knew of doing that was giving some things that could work, rather than focusing on the things that are highly unlikely to.
If you do decide you want help in developing your college list in preparation for your visit with your high school guidance counselor, let us know more about what it is you’re looking for in a college.
Wishing you the best.