Hi Thy Apathy,
I am a recent graduate and math major from Amherst College. I was actually admitted to both Princeton and Amherst but ended up choosing the latter over the former. I was pretty put off by the eating clubs at Princeton, since these seemed to be nothing more than institutionalized cliques. More than anything, though, I simply like the atmosphere at Amherst better when I visited, so I will echo others in encouraging you to visit schools to which you are admitted.
Things I liked about Amherst:
Small faculty
There is a very tight-knit math community at Amherst. Professors are almost always in their offices and welcome unscheduled drop-ins. What's more, the faculty is top knotch. Dan Velleman's How To Prove It is one of the most-used introductions to proofs and David Cox has a very good book on Galois Theory. And then there's the ageless Norton Starr, who gave the best Calc II lectures I've ever seen and offered chocolate to anyone who could catch him making a mistake at the board. As a math major, you are both invited to and expected to attend the regular talks hosted in the afternoons and the weekly math table was always a fun, nerdy lunch.
The Five College Consortium
As a student at Amherst, Hampshire, Mt. Holyoke, UMass, and Smith, you can take classes at any of the aforementioned schools. As n1person pointed out, the small size of Amherst's faculty reduces the number and breadth of courses offered, but with the addition of any and all math courses at the other 4 schools, I think it's safe to say the selection is even broader than the average university. Two other math majors from my year took grad level classes at UMass in algebraic topology and differential manifolds. There are even students who have completed honors theses with advisors from other schools. There are also many collaborations between 5 College professors and joint conferences as well (e.g.,
http://www.math.umass.edu/~siman/seminar.html).
n1person mentioned there are many students at Amherst who have math as a secondary major, but I hardly counted them as a part of the math community. If I'm not mistaken, the completion of high level math courses is a requirement for admission to grad school in economics and physics and I imagine you might find a similar percentage of math-______ double majors at other schools.
I have no idea what the admission process is like, and I don't think anyone does who hasn't actually worked in the admissions office. There was an enormous variety of different types of people at Amherst, and I sincerely doubt that there is some "formula" for how to become the perfect applicant. I had very good SAT scores and grades and took several AP classes, but I came from a tiny high school in the sticks.
Frankly, if you're looking for more information on any department in any school, I think you should simply email one of the professors with your questions.