How much does my undergraduate school matter for graduate admissions?
This is a tricky question to answer, and opinions are going to vary a bit. The general consensus seems to be “Some, maybe a lot, but what you do in undergrad is way more important than where you go.”
Let’s be honest - professors, like most people, can be swayed by prestige. But it’s not the sparkliness of the name; it’s the familiarity with the department, the faculty, and the work and rigor that go into that program. If your field has excellent departments at Wisconsin and Duke and Michigan and Stanford, then if you come out of one of those universities the professors at your graduate schools know that you had good training at the department. They know the faculty members there - maybe they went to grad school together, or collaborated on past projects, or were postdocs together - and they trust their word in letters of recommendation. They know that the cutting-edge research is coming out of so-and-so’s lab, which just happens to be where you did your RA job. So when your application crosses their desk, a lot of the things you discuss are known quantities.
However, this does not mean that if you go to East Carolina University, Cal State Northridge, UNC-Wilmington or Loyola Marymount that you have worse choices of getting into graduate school. This also does not mean that “a 3.3 at Stanford is the same/better than a 3.7 at CSUN.” This also doesn’t mean that you should go deep into debt to attend a more prestigious school, or transfer away from your current undergrad. It’s simply a data point - one that’s taken into consideration. There are lots and lots of people who go to excellent graduate schools from these schools and other smaller regional publics and lesser-known privates. What’s more important is what you do. So get involved in research (yes, there is research going on at places like these), form relationships with professors (yes, they still count even if they don’t know Professor Fancypants at Harvard), try to do a summer research internship at a different university, and take the most rigorous courseload you can (consider taking graduate courses if you are able).
^From the FAQ.
For example, would a 3.3 at, say, Vanderbilt be more likely to get into a top grad school than a 3.7 at UNC Chapel Hill, all other things held equal?
UNC Chapel Hill is an excellent school, and I don’t think graduate school admissions officers would think much of an academic difference between UNC and Vanderbilt. UNC and Vanderbilt are also roughly equivalent in terms of research and excellent departments - in fact, I would say that in most fields that I am familiar with, UNC has better departments. So a Vanderbilt degree would not necessarily impress an admissions committee more than a UNC degree.