It really depends on what type of graduate school you want to do. Do you want to go onto a PhD program or a Masters program. How competitive is your field?
It’s unfortunate, but the reputation of your undergrad school does matter a lot in terms of getting into graduate school. This is for two reasons:
-
The students you are surrounded, the degree of competition, and the quality of teaching and faculty. Your performance is affected by your environment and what is expected of you. You want to be pushed as hard as possible, without breaking. These factors also affects how professors can measure and stack you against others. Are you the best in your program, in the top 10%, etc. Being an excellent student at UW-Madison or Minnesota would be perceived as much stronger than being the top student a professor has taught for many years at a random school like UW La Crosse.
-
You want your recommendation letter writers to be known to the graduate schools where you’re applying. Professors in many departments at UW and UMN are world renowned and if they write you a strong letter of recommendation, then you are basically set for whatever grad school you want. Professors at well known LACs are also well known.
If you’re applying for a not so competitive field like social work, teaching or nursing, going to the satellite state school should not affect your chances so much. Also, consider your future expected earnings when deciding how much debt you want to take on.