R127, Rutgers' full name is Rutgers, the State University of New Jersey. It also has three campuses, each with different emphases. Elizabeth Warren went to its law school, where Ruth Bader Ginsberg also once taught.
The Ivy League universities each have their strengths and weaknesses, but on the whole they are among the best universities not just in the US but the globe. Harvard, Princeton, Yale, Columbia, and Penn are all very strong in most areas, and Brown, Cornell and Dartmouth are also very good, if not as highly ranked. They also vary in size; Cornell is huge, Harvard and Yale contain all kinds of schools (Forestry, Divinity, etc.), and Princeton is the smallest in terms of divisions and offerings. It is also the richest per capita.
There are other private universities that as highly ranked: Stanford (the premier university on the West Coast and one of the best in the world), MIT (the leading science and engineering university in the US), the University of Chicago, Duke, Northwestern, Johns Hopkins, California Institute of Technology, Washington University in St. Louis, Rice University, University of Notre Dame, Vanderbilt University, etc. You can Google this, but these are all among the major research (Research I) universities in the US and globally.
The best public universities in the US are: UCLA, the University of California-Berkeley, University of Michigan, Wake Forest (it's very good), University of Virginia, University of Texas, Georgia Tech, UNC, UC Santa Barbara, and UC Irvine. Not sure about how much credence to give a term like "Public Ivies," but it's the case that there are many very good public universities, and if you live in state and pay state tuition, you can probably get a superb education for 1/2 to 1/3rd of the price of private universities, though some, like Harvard, do heavily subsidize students whose parents make under a certain cutoff.