So, how much does your undergraduate institution matter in grad admissions?

“Graduate admissions” is a huge category. Ph.D.? MD? JD?

On an extremely general level, though, the school is less important than your performance both in and beyond the school. You’d need excellent grades (3.7+) and then something distinguishing yourself beyond the grades (research, work in the field, etc.). That and great GRE/ MCAT/ LSAT/ GMAT score (and by great I mean in the high 90th percentile for a competitive grad program).

Law schools tend to accept more by numbers than other things.
I hear on this forum that med schools are similar: numbers first (grades and test scores) and some clinical experience
Ph.D. programs each will have their own criteria and often individual professors will accept you, or not. Each of their sets of criteria will differ.

Get great grades. Do good research. Fr Ph.D. research programs, if you are forming ideas, scholarly ideas, so much the better. If you can publish even better. Do good practical work in your field. Practice the entrance exams until you could teach the entrance exam to others. Then you should be a good candidate.