Most of the best people that I have worked with over the years went to either a state university, or a university that I had never heard of. The best software engineer that I ever worked with went to a state school. One of the best bosses that I ever had went to seminary school (which was also one that I had never heard of). The worst software engineer that I ever worked with had graduated from Harvard (I will admit that the fact that he had graduated from Harvard might have slowed down his being fired).
The stigma around non-prestigious schools pretty much starts and stops with high school seniors, investment bankers, and maybe expensive management consultants. If you intend to be neither of these in your professional life, then you can go to any university which seems like a good match to you. If you intend to be an investment banker or management consultant where you get your Masters or PhD will matter, where you get your undergrad will not.
“I’d like to add that I am planning on going to medical school after undergrad”
Medical schools care about undergraduate GPA, and MCAT scores. If you go to MIT or U.Chicago, getting a great GPA is going to be more difficult, which will hurt your changes of medical school. Of course MIT and Chicago attract a lot of very strong students, and premed students from there do often get to medical school. However, if the same student who went to MIT to study premed had gone to UMass Amherst instead, she or he probably would have had a higher GPA and the same MCAT and done just as well or better in terms of getting into medical schools. He or she might have had less debt also, which would help considerably in the long run.
Also note that there are a LOT of very strong universities in the US. Any school in the top 200 is very good. No school is a good match for the majority of students, but every school at least in the top 200 (and probably in the top 2000 or more) is a very good match for some students.
GATech and UofRochester are very strong schools. As others have pointed out they might not be truly safeties, but they most certainly would not be considered anything other than a strong university if you were to graduate from either. The same could be said about hundreds of other universities.
“I feel like I will be shamed when talking to others”
I mentioned UMass above. Let’s suppose that you are from Massachusetts, have straight A’s (mostly A+'s), and go to UMass. Some kids in high school might bug you. As soon as you show up on the UMass campus you will be surrounded by other kids who also chose to be there. With straight A’s you will probably be in the honors college. You will get a great education. In four years you will graduate with a degree from a great school, and less debt that you would have had elsewhere, and employers will be interested in what you can do for them. If you go into management, in a few years you will probably have a few MIT and Harvard graduates working for you. They won’t care where you went to university.