Generally getting into a program such as Harvard/Princeton/CalTech is going to be extremely difficult for most people, their acceptance rates are so incredibly low that getting in is tough.
Don't let this discourage you tough, a lot of great minds come from all over, from many backgrounds, and the education you would receive at a quality state school or smaller private can be just as good, you just need to be certain you bust you butt and achieve academic excellance in undergrad.
Grad school is considerably more important than undergrad school, so I would visit each school and make a personal decision about where you want to go... don't let the USnews rankings or some other BS skew your desicison too much... you want to go where you will be HAPPY, not just the most "prestigious" place you can get into...
Also, many such top tier programs are ranked as such for their grad school programs, while the quality of their undergraduate programs can vary widely... I suggest sites like collegeconfidential.com and others to get a feel