Sorry to burst your bubble, but there is a pretty standard procedure for getting into law school.
They look at your LSATs and your GPA, but they weigh the prestige and difficulty of your undergrad's institution.
This isn't a "look at us we're so fancy here at Cornell" type post. Yes, Cornell is very respected by law schools. But there are several schools that technically rank better.
The point I'm trying to make is that it would be slightly easier to get into a top law school from Cornell than NYU, and much easier than from Binghamton.
A bit of anecdotal evidence here: I have met so many people that got into top 10 law schools from Cornell that I'm surprised when I hear someone didn't get into one of these schools. Take that for what it's worth.
There are also several classes taught in the law school that are open to undergrads that are very popular - not to mention pre-law fraternities and relevant organizations to fine tune your ambitions with likeminded students and faculty. I took several classes as an ILR'ie that would have doubled as law school classes somewhere else. They completely sucked and I hated them haha. Shout out to Professor Gold, that sack of shit. That class was so pre-law, it convinced me that I didn't want to go to law school!
I'd say for your purposes, you should go to either NYU or Cornell. And pick a major that will gear you towards law school.
ILR is the probably the biggest pre-law major at Cornell. Throwing out a number here, but I'd say at least 80% of ILR'ies consider going to law school at some point. Government, psychology, and PAM are also fairly common.
I really hate to be "that guy," but I'm going to warn you against philosophy. Take some philosophy classes, but don't major in it. It's a recipe for disaster. Just in case law school doesn't work out right away, it's better to have a major you can actually get a job with. (And I say this as a resent alum who has had plenty of experience with the real world job market). It would also help you to have some practical knowledge and tangible skills while entering law school rather than a, dare I say it, (cough useless philosophy degree cough cough).
Best wishes and good luck with your decision.