In need of advice.

I just graduated high school with a 4.0 GPA, 10th out of 219, with a 1560 SAT and president of almost every club I joined.

I applied to 9 schools. I got into my safety, wait listed to three (NYU, GWU, UChicago) but rejected from the rest which were Ivy Leagues (Harvard, Yale, Stanford, Columbia, Princeton).

I want to know if I should attend my safety, (Rutgers), all four years on a pre-law track and apply to Harvard law. Or should I try to transfer to an Ivy League within my first two years? In either case I would like to end up going to Harvard Law. Which path is most recommended?

Was that 1560 out of 1600 or 2400?

Sounds like you are putting the cart before the horse. If you haven’t even started school at Rutgers yet it does not seem like you could make an educated, wise decision about transferring (to an Ivy or anywhere else).
The path that is recommended is the one where you will be happy, learn a lot and have significant personal growth.
If you are interested in law that is terrific. Harvard Law is certainly a laudable goal but first take some classes and see how that goes.

Could you afford Harvard even if you could get in as a transfer?

If Harvard law is ultimately your goal, then just go to Rutgers, get a near perfect GPA and score very high on the LSAT, that is what most law schools want.
One thing for sure: Where you went for undergrad is not a major factor in law school admissions.

Stay at Rutgers for all four years. Get the best GPA you can and develop close ties with faculty members who can write the kind of letters of recommendation that make law schools want to admit you. And save as much money as you can because law school is expensive.

Your other option, if you are not happy going to Rutgers, is to take a gap year, intern in a legal aid or similar organization, and reapply to a different range of schools next year. Starting at a school with the idea that you aren’t going to stay is a recipe for a poor experience.