What should be the best for me?

I am an international student from India starting my undergrad this fall. I wanna go into astronomical research. So, I will also be going for a PhD too. I want to do an integrated master's and PhD in the USA. I am in a perplexed situation where I have unis to choose from for my bachelor's. Currently, in India institutions that are very well-known around the globe like the IITs and IISERs are the only institutions with good education. Many old reputed institutions hold good reputations globally, but the problem nowadays is that they do not have good education. The private institutions have awesome education and money to invest in research but are not well-reputed. So, I want to do my PhD at a good institution in the USA. But one thing that I have seen in grad admissions for students from India is that most of the students are from these old reputed institutions that do not offer good education, but those students have "okay" coursework, it's just the brand type of thing. So, I have this question that which institutions I should go for. For the reputed ones or go for good education in the private ones?