When choosing a college, I found that I had an abundance of resources to help me. Every school in the country sent me brochures, in English class we wrote essays for the applications, I got books and magazines with lots of information about lots of schools, and everyone else around me was making the same decision.

Applying to grad school was a completely different experience. Most of my friends aren't math majors, and even those who are were mostly looking for jobs. I had no clue what a personal statement was. Nobody sent me information until I asked for it (and sometimes not even then), and nowhere on the web could I find a page of helpful tips for applying to grad school. So now I'm trying to correct the last situation. Note that all of this advice is intended for people applying to grad school in math, with the intention of getting a PhD and becoming a professor. It may be applicable to others, but I make no guarantees.

• Start thinking about grad school before you leave at the end of your junior year.
• Talk to professors and grad students. They are your best resource, and all of them should be happy to help you. They went through the same thing when they were undergrads (yes, even 74 year old professors were once undergrads).
• Request applications over the summer. It's usually quite painless; there's either an online form or someone you can email (just say "Hi, I am a senior majoring in math at . I plan to enter grad school in the fall of 2000 to pursue a PhD in math. Please send me an application and any relevant informational materials. My address is: . Thanks, ").
• Some schools are using online applications. I found these worked fine, and save a bit on postage (plus you can pay the application fee with a credit card). It is still nice to have written material, though, so go ahead and request the paper forms and brochures.
• Talk to people from/at the schools you are considering, especially if any of them know you and you interests. If everybody is unhappy there, then don't waste your money applying.
• Apply to schools you would consider attending. If you know what field you might want to work in, then you can even think about people you might want to work with and see where they are. For the most part, however, even if you've taken a bunch of classes, you probably aren't very sure what you want to do, and should apply to schools that have good people in lots of fields. I applied to 5 schools, and that felt like a good number.
• Why apply to Harvard? They have the earliest deadline, and it costs so much that you could apply to two other schools instead. Unless you're sure you want to go there (talk to current grad students there to get an impression), don't bother.
• The US News & World Report rankings are nice to be aware of, but don't take them very seriously. They do mean a lot more than their undergrad rankings, though, which really should be ignored.
• Apply for an NSF fellowship. It doesn't cost anything to apply, and most schools will offer you more money for less work if you have NSF money. You can apply online, so it is quite painless. I've heard they put a lot of stock in GRE scores and past research.
• The Department Of Defense also offers a good fellowship that's worth looking into. There are other good fellowships to consider depending on what you're studying, such as the Hertz fellowship in "applied physical science."
• Note that you do want to get some teaching experience even if your fellowship means that you don't have to. What are you planning to do after getting your PhD, after all?
• Take the GRE's whenever you feel like it. The general test can be taken at any time since it's done on computer, you just have to make an appointment a few weeks in advance. I suggest taking it early, such as around the first week of classes senior year, just to get it over with when you don't have much else to worry about. The subject test is on specific dates, and you want to sign up very early to get the testing location you want. Note that if you're applying for an NSF fellowship, then they will pay for your subject test if you take it in December. I didn't find out about this until I'd already paid for an earlier test date, but this can really save you money.
• I heard two basic types of advice for the personal statement (and the expanded version of it for the NSF application). Some people said they picked a random topic and wrote about how they really really wanted to do research in that field and work on that problem. This is complete bs, of course, but apparently it sometimes works. If you know you want to work in a certain field, this could work for you as long as you don't claim to have a thesis in mind already. The other approach is to be honest, say you aren't sure what you want to do, but talk about the math you've enjoyed so far, and how you really want to learn a bit more, then settle on a field with an advisor. This worked fine for me.
• Get everything in well before the deadline, or as early as possible. The most difficult part of the application is getting your absent minded professors to actually write letters of recommendation on time. I found it convenient to put all the material, including the letters, in one envelope so as to minimize the chance of anything getting lost.
• Ask for the letters of recommendation way in advance, i.e. by November 1. As soon as you know what schools you are applying to, put all the forms (with envelopes) for each professor in a big envelope, and write the contents and due date on the cover. Remember to waive your right to see the recommendations. You'd only be able to see them if you get in, and at that point why do you care what they said? Send periodic reminders until the letters are done. As the last minute approaches, find the professor and stay with them until they finish with your letters.
• You want 3 professors to write letters of recommendation for everything, and for the NSF fellowship you should ask a fourth professor to write one.
• Don't be afraid to ask if someone will write you a letter, and if it will be a good one. You want the best possible letters in your application, since they are the most important part of the application.

• Some schools will accept students as early as late January, others will wait until mid-March, so don't expect to hear from everyone around the same time.
• If anyone wants your reponse earlier than April 15, have the head of your math department kick some ass for you. This is a very difficult decision, and nobody should be allowed to pressure you.
• If you aren't guaranteed at least four years of funding, then don't even think about going. You should expect to have your tuition paid and be given a stipend that will be (barely) enough to live on. Note that some schools are far more generous with money than others. You don't go to grad school to get rich, though, so try not to let a few thousand dollars make the decision for you (yes, it is very hard to turn down a school that is offering you $6k/year more than the one you want, but you have to focus on why you're going to grad school).
• Go visit. Ask how much they will pay for you to visit, and expect an answer of $400 (unless they're close, in which case you should expect a much smaller number). You may have to shop around for a cheap plane ticket, and take a bus to a distant airport, but it is possible to travel cross-country for under $400. You definitely want at least one full weekday there, but it's not uncommon to take a Saturday night stay to save money.
• Note that Stanford is not very good with reimbursements. I visited at the beginning of April, and it wasn't until October that they could even figure out that they'd lost my paperwork. If you're short on money, then keep that in mind.
• I found it very useful to take a light load in the second term of my senior year. For one thing, grad school decisions are made with no knowledge of what you are taking. Also, it makes it far less stressful to take time off to visit schools. Getting a tuition rebate is also pretty cool.
• Trust the grad students. That's not to say that you shouldn't trust the professors, but they are trying to recruit you, while the grad students have no incentive to lie.
• Attend a class just to see how it compares with what you're used to (does the professor respect the students, are the lectures interactive, do people look interested or lost?), but spend most of your time talking to people and trying to get a feel for life outside the classroom. After all, the primary purpose of going to grad school is to specialize in a field and write a thesis, not to sit in lectures.
• Ask people at each school what they think of the other schools you are considering in comparison, but be wary if they quickly say something negative about another school (if you visit the University of Chicago, you'll find out who I'm talking about). If they praise both schools and point out relative strengths, then they're probably being honest. If they immediately say something bad about the other school, then question their motives. All schools are good for some people, otherwise they wouldn't exist. What you really need to figure out is which one is best for you. Sometimes the quick negative answer really is honest and appropriate for you, but don't let a single conversation dominate your opinion of a school.
• I found it funny that you're not expected to know what you want to do when you go to grad school, yet the first question everyone asks is "What do you want to work on?" I never did figure out the best way to reply to this, but some people suggested that simply "I don't know" often works well. I think the most important thing is to be perfectly honest, with others and with yourself.
• I asked almost every grad student I talked to for their favorite and least favorite things about being a grad student at their school. This seemed like a useful question.
• Don't give up if you're waitlisted. Ask what your chances are of getting in, and if it's really the school you want to attend, then tell them that and ask if you can be first on the list. It is not uncommon to go to a school where you were waitlisted.
• Take your time deciding, but notify schools you've decided against as quickly as possible. This is polite, and it makes life so much better for the person they then admit off the waitlist. Note that your decision is not final until April 15 (and it's not impossible to get an extension, if you really need it). I met several people who ended up going to schools they had originally declined.
• Conventional wisdom is that you should go someplace new for grad school. It is not necessarily bad to stay in the same place, as long as you know why you're doing it ("just 'cause" is not a good reason). The most amusing reassurance I got was "Just because you made a mistake by coming here as an undergrad doesn't mean that you should make another mistake by leaving for grad school."
• Whatever schools you are considering, you will hear both good and bad things about them. Schools can change in just a few years, so some people's opinions might be outdated. Current grad students have the most current information, but also can't give you the outside perspective that a professor from another school can. Gather as many impressions as you can, because you really want to make an informed decision. After April 15, you will suddenly rediscover what it means to get a good night's sleep.