Good question. There are many colleges/universities in the US, something like 3000. They are wildly different.
A few main points:
Public and Private: Some are public (funded by state governments) and some are private. A few dozen of the most famous schools (Harvard, Yale, Stanford, Chicago, Johns Hopkins) are private. However, you can’t use public v. private to determine quality. Many of the public schools–Universities of Michigan, Wisconsin, California, Texas, Virginia, North Carolina, William and Mary, etc.–are some of the best universities in the world.
Public and Private (costs): Privates typically begin with the same price for all students, domestic and international. Publics are cheaper for in-state students because their families are paying that’s state’s taxes and thus paying for the school. They cost more for out-of-state (OOS) and international students. Public universities typically give less financial aid to OOS students. Some will give merit aid to very top students. Your best bet for aid will be private schools, and it is more difficult for international students to qualify for aid. If you are full pay, then definitely look at the publics.
Size: You also cannot determine quality by size. Some of the large state universities have up to approximately 40,000 undergraduate students. Some of the best liberal arts colleges (LACs) have 2000 or fewer students. The general rule of thumb is that these large state research universities offer better research opportunities and the smaller schools offer smaller classes (often much smaller) and closer interaction with professors. But you can get great research opportunities at smaller schools and close interaction with professors at large schools. That’s often up to the student to make those things happen. I’d call under 4000 small, 4000-10,000 medium, 10,000-20,000 large, 20,000+ very large.
Location: The Northeast has many great schools, including lots of top LACs. It is more liberal politically and has very cold weather. The Mid-Atlantic (about PA to NC) also has many great schools and better weather. The South is more politically conservative (although campuses are less so). The Midwest has many good schools, cold weather, and maybe the best deals (merit aid/financial aid). The West generally has nice weather (can vary). There are fewer schools, mostly because many of the mountain states have small populations, and California has many excellent public universities, though good privates as well. Which region sounds best?
Environment: Would you like to be in a city, a college town, or a rural area?
Once you think about these, you can go to a college search site like Forbes, US News and World Report College Rankings (USNWR divides schools into universities and LACs), or College Niche to find schools that meet your most important criteria. You can also google by almost any category: Best large universities, best southern colleges, best urban colleges/universities, even best hiking colleges.
Also, once you have some parameters, you can also ask people here on CC what schools would fit what you want: for example, what’s a good urban school in the Midwest or Northeast for an international student interested in chemical engineering who has x test scores and x grades can pay $30,000 per year, OR what’s a good school for an international student interested in studying business who has x test scores and x grades and will be full pay? OR what’s a good school for an international who has x test scores and x grades and needs significant financial aid? Good luck!