These are some pieces of advice and reflections after my time here at Cornell.
Disclaimer: As with all advice, of course different people have different needs. So I'll state my biases up front: I came into college with a strong background in engineering-type math (I had taken multivariable calc my senior year of high school), but almost no exposure to proofs outside of a couple of discussions with my HS math club. I now have interests in algebra and computer science, and am going the pure-math-grad-school route.
We have fantastic library resources! Almost any book or article you could possibly want is available online. Just search through the main libraries page.
Go to office hours! Even if a professor is not the best lecturer in front of a full class, they are almost always much better 1 on 1 or in small groups as with office hours. This is a resource I regret not taking more advantage of. If you think you fully understand the material, you should go to try to get a broader perspective or hear about related topics. If you're totally lost, this is the best way to get unlost—even if you're so confused you don't know what you're confused about, it's fine to start with a vague question and narrow it down as you hear more. In the long term, this is also a great way to meet professors, since at some point you'll likely need rec letters or references.
Too shy? Get a friend or two to agree to go with you (this is what I've done almost every time the past 4 years!) Or try to plan out questions in advance. Good ones are "Could you give me more examples of X?" or "I didn't quite understand the definition of Y in class, can we go over it again?" or even "I'm really confused about broad area Z, could you just tell me more about it?".
From Oliver: Ask questions in class as well! It will also help the professors get to know you better. Plus, you're probably not the only confused person, so you're likely doing everyone else a favour.
If you have an interest outside of math, try to explore it. For me, this is music: even though I've been in an ensemble all 4 years, I've only taken one music class. I wish I had taken more, since I'm starting to realize that I might not have as many opportunities going forwards (how many places could I find a gamelan or steel drum ensemble? when will I have time to learn electronic composition?). This is also a good place to take advantage of taking courses S/U, so you only need to spend as much time as you want, or trying to strategically fill distribution requirements only using your interests.
I've learned a lot more by working with friends. I get to hear different perspectives and approaches to solving problems. And explaining things in return, and talking through my own confusion, helps solidify my understanding. Of course, not everyone has compatible working styles (even if you're otherwise friends), so keep trying different people until you find some who click.
From Shruthi and Oliver: be open to learning different kinds of math. And even if there is a kind of math that you don't like doing, try and appreciate its benefits and the people who do work on it.
From Oliver: take grad classes which actually have homework, or you won't learn anything.
Learn to tex. You can't get a dog until you get tenured. For talking to profs: you're not expected to remember everything, so it's ok to ask the same question more than once. You don't need to take a point-set topology class, you can learn it as you go along. Some textbooks are bad and some textbooks are good but have mistakes. Some mathoverflow and stackexchange posts are just wrong. Get to know your advisor. Get some sort of tutoring or teaching job&emdash;it's good to get experience explaining math to other people. Don't do things just because they sound fancy.
For applying to grad school: look at what alumni of certain advisors are doing. This can give you a better idea of what you'll be doing than the advisors own research.
From Oliver: don't be too specific in the kind of math you do since you're not sure what the reader's background will be. And don't get too attached to what you did as an undergrad. There's a kind of expectation that your interests will change when you get to grad school.
You'll have to take the math subject test as well as the general; the math is by far the more important one. They're quite strict about all the various requirements for test taking, so be sure to read them carefully. For example, you need a certain kind of valid photo id, and when I took the general they even inspected my glasses to make sure I wasn't hiding any notes. Test times can also fill up quickly, and you have to sign up at least a month in advance, so really sign up as soon as you know for sure you're taking it. Both are unfortunately expensive, so make sure to budget for it and sending scores. Definitely take advantage of the free score send! Even if you have no idea which schools you want to apply to, there's absolutely no harm in sending a free score to a school, then not actually applying there. So take your best guess at some schools and use the free send! Caveat: if the score is bad, the schools now have a bad score. So perhaps pick some safeties, or hold off if you're really unsure about how you'll do.
The general GRE is the easier of the two (though obviously for non-native speakers there is a bit more challenge). When you sign up, you get access to a couple online practice tests; go through these so you're familiar with the format (after you've read the descriptions of the section on the website). Try to take it when you're at home, because the only options around Cornell are going to Binghampton or Syracuse. There are three types of sections on it.
Subject tests are only offered 3 or 4 times a year. If you can, take the math one in April of the year before you apply (or even earlier if you want). This way, if you do badly, you get your score back in time to know whether you need to take it again, and if you do well, you don't have to worry about doing it on top of every other application thing in the Fall semester. The other test dates for subject tests are some subset of September, October, November; if you need to take one of these, do the earliest one possible. The subject will actually require some studying, since it's a big mix of subjects. Most of it (i.e. about 50%) is calculus, so that's the most important section to have down.