First of all, with an intention to major in neuroscience, it is a very reasonable plan to intend to go on to graduate school and to keep this in mind. This is a major where some form of graduate school is very possible, and probably likely if you stay in neuroscience. There are of course lots of other related biomedical fields where some form of graduate school is also possible. I have several family members who majored in some biomedical field, and the number of graduate degrees matches (one graduate degree is still in progress).
There are a lot of universities which are very good for a pretty wide range of biomedical fields, including neuroscience.
If you look at the various graduate students at a highly ranked university in pretty nearly any highly ranked graduate program in biomedical sciences, or psychology, or neuroscience, they will have come from a huge range of undergraduate programs. A few will have come from Harvard or Columbia or JHU or NEU or somewhere similarly ranked, but most will have come from a huge range of other schools. “All over the place” is one way this has been described.
What will these graduate students have in common? Several things. One is that they did very well as undergraduate students, and had shown up at university on day 1 ready to work hard and take all of their classes seriously. You can do this at any university. They will have found good research or internship or other related experience. You can do this at a very wide range of universities. They want to do it. Getting a PhD is for example a rather long path that takes quite a bit of determination.
Another thing that they will have in common is that they can afford to do it. If you take on a large debt to get a bachelor’s degree this can make it difficult to afford to attend graduate school.
Looking at the students at a particular highly ranked biomedical PhD program, I see quite a few that have a master’s degree, but probably just as many or perhaps slightly more that do not have a master’s degree. You can in some cases go straight from a bachelor’s to a PhD. You will need very good grades, very good references, very good research experience, and a good explanation of what you want to do and why, but again students from a very wide range of undergraduate universities manage to pull this off. One big advantage of skipping the master’s degree is that a PhD in a STEM field is usually fully funded, whereas a master’s degree is not. Thus a two year master’s degree might very well cost a student something like $200,000 (and that is today, five or six years from now it will be more expensive). For a student in a PhD program, the university will often (usually?) pay your tuition and fees, provide health insurance, and give the student a stipend that is enough to live on provided that you live very frugally. What this all means is that if you can get through a bachelor’s degree with no debt, and with good results (such as grades and research experience) then while it is a very long and challenging road to get a doctorate, it can at least be financially possible.
I by the way would not allow my daughters to take on any debt at all for their bachelor’s degree. One daughter was frustrated by this. She got into NEU and BU with offers that were not affordable. She went somewhere else. She got a great education (and two bachelor’s degrees, one of which was in neuroscience). She got a dream job, but a dream job that paid badly. It could pay badly because it was that attractive as a job. She could only take the job because she had no debt. This job led to a second job, which provided experience and references that helped her a lot in her graduate school applications. At one point she was frustrated that I would not let her take on debt for her bachelor’s degree. Five years later she thanked me (twice) for not allowing her to take on any debt. She is very much aware today how much better off she is because she graduated with no undergraduate debt.
You should find out from your parents what they can afford without taking on any debt. Then you might want to have your parents run the NPCs for a few schools on your list and see what they are likely to cost, if you get accepted.
You should make sure that you apply to at least one university, and even better two universities, that are very, very likely for admissions and also very, very likely to end up being affordable without taking on debt.
Earlier you mentioned that you would like to be in the northeast (which is where I live and where one daughter is currently studying for a PhD). Certainly apply to universities up here. See if you get in and see whether they will be affordable. I am just trying to encourage you to make sure that somewhere around about 12 or 14 months from now you will have at least one very good affordable option to choose from. Given how excellent the Ohio State University is, and given that you are in-state, I think that you should apply. That does not mean that you will go there. It means that you are that much more likely to have at least one affordable option at a very good university when you are looking at offers in about 14 months. Then you can think about which of multiple offers to accept.
And if your goal is to get to a graduate program at Harvard, or BU, or NEU, or Tufts, or JHU, or Columbia, or at some other excellent program up here in the northeast, any of the universities that have been discussed on this thread can provide the undergraduate education that will help you get there. Whether you can get accepted, and whether you can graduate with a bachelor’s degree and still be in a position to afford to attend graduate school, might vary depending upon which schools you apply to.