“What kind of college or university should I go to?”
In the United States, there are many colleges and universities and they are of different kinds. It is worth knowing about this before considering which schools to apply to.
First, there is the research university, which is often large and has graduate students, including at the doctoral level. And there is the "liberal arts college," which typically has only undergraduates and is relatively small.
There are some universities that combine the two: Most of the Ivy League Universities in the Northeast, including Harvard, Yale, Princeton, and Columbia; Stanford; University of Chicago; University of Southern California are among them.
The state universities are all research universities, more or less.
The liberal arts college is devoted primarily to teaching. The faculty are scholars and do contribute to research and publishing in their fields, but they do not have graduate students and typically classes are small seminars rather than large lecture hall events supplemented by small seminars led by a graduate student and sometimes called "discussion sections."
In a lecture hall, the professor speaks, the students listen and typically take notes, and the professor may allow questions towards the end of the session. In a seminar, the professor leads a discussion in which everyone in the room can participate. There is at least one other style of teaching and learning: the tutorial. A student meets with the professor one on one for a discussion between them. For that, one more or less needs to go to Oxford or Cambridge in England. Our universities sometimes have a bit of this in "office hours." These are usually voluntary. You should make use of them. Both because of the additional learning opportunity this provides, and because it gives you an opportunity for the professor to get to know you, and that will be handy if you want her or him to write you a letter of recommendation (or read your paper, perhaps with the hope that he or she might).
The most famous professors who are doing cutting-edge scholarship in their field (and this can be exciting to discover, and is surely one of the great things about the better American universities) are usually at large research universities. One advantage of being an undergraduate at one of these schools is that you can take courses with and maybe even get to know (and also maybe get a letter of recommendation from) one of these superstars. If you are drawn to study at one of these great universities because you get off on interesting ideas, then you are not alone, and this could be the right place for you.
But you will get less attention, and you will have to clamour and jockey for the attention of one of these star professors. This is not a problem generally for graduate students, at least not in the same way, and not in PhD programs. Most graduate courses are seminars. And they know you are very smart because they admitted you to their program, whereas at large universities undergraduates tend to be treated like they may or may not be one of the very best students and future scholars.
One more thing: the better American colleges and universities not only train and socialize students for elite professional jobs (or to be able to compete for them, and note that this is why colleges recruit "leaders" and that is so big a word); they also have a plethora of student services of all kinds. You pay for these things, and European universities that are free or inexpensive as many are (in some countries, even to foreigners, though you may have to have learned the language rather well first) tend to have large lecture hall classes and few other services. That is part of what you are going into massive debt for, as well as to have great professors, lots of personal attention (the American college, again), or both. You also get to spend much of your time, especially in your first two years, taking courses in various fields, following your interests, delaying commitment to a major course of study, or taking courses with famous professors in other departments. Many Americans from the middle class becomes extremely curious about some or many things for four years. Make of it what you will.