The graduate school application process is essentially over and my results were pretty good, so I will put everything that I think was useful here. **This will be tailored to a research-oriented graduate school application rather than, for example, a medical school application. They require completely different approaches.**
Let’s think about that in terms of each component of the application: letters of recommendation, statement of purpose/research experience, and transcript/GREs.
Letters of Recommendation
This is one of the most important parts of the application, and for good reason. One common problem among people is that we have trouble accurately judging our own skills. We can say that we are on time, good with people, and so on, but it could be way off. Outsiders often have a better objective view of how we work. Therefore, the letters serve as a sort of litmus test. If your previous professors enjoyed working with you, then there is a good chance that your grad school advisor will too. This idea also explains the tier list for letter writers (professors > industrial researchers > bosses): the closer the experience is to research, the more likely it is that the results will be similar. For example, research in industry is generally less open-ended than in academia, and so someone might be great when there are step-by-step deadlines but struggle with few constraints. That is less likely if you have already had some success in academic research. Probably the best ways to get good recommendations are to take initiative, be interested, follow through with what you say you will, and generally be pleasant to work with (this last one goes a long way wherever you are).
Statement of Purpose
There are three keys to doing an excellent statement of purpose: (1) demonstrate that ***you know what you are getting in to***, (2) demonstrate interest in the specific field or subfield that you are applying for (past research in the area is great for this), and (3) try to show some capacity for self-learning and motivation to deeply understand and publish new insights on a topic.
A few things to avoid: (1) do not talk about your childhood dreams and (2) do not talk about your outside interests. Childhood dreams are an easy (and common) way to start the statement, but admissions committees do not really care. They want to know about who you are right now. Specifically, talk about research and demonstrate passion and a realistic picture of what graduate school will be like, because this is really a research statement rather than a personal statement. They want to see a well thought out topic and purpose for applying to graduate school.
Here is how I did it: I started off by saying that I do not enjoy classes and instead prefer to teach myself. (This suggests that I am comfortable with the self-guided learning that is important in research.) Next, I mentioned the field that I was interested in and why I am interested. (This helps them direct the application to the best professor to read it, and it is a good place to demonstrate passion. Put the field in bold for their convenience, like “… and for that reason I want to work in nanotechnology and energy.”) I also mentioned other semi-relevant interests in one sentence here: “I am also very interested in business and education, and I plan to make each a significant component of my graduate experience.” (It is useful to mention your other interests because people with a variety of interests tend to be more interesting people and they may identify with yours, but do not waste any space explaining them. I literally only mentioned business and education in the one sentence above.) All of that took a little over a half a page, and I spent the next 1.5 pages talking about my past research and engineering experience. It is good to mention what you did, what went well, what you struggled with, and how you improved as a researcher from the experiences. Also talk about the field you want to go into in more depth. Talk about the problems you find interesting and about your ideas and questions regarding those subjects. I spent about a paragraph and a half on that. I finished up with talking briefly about my professional goals and how the school is an excellent fit for both my field of interest and my goals. Your chances of admission go way up if you fit with the institutional culture. For example, if you are applying to Stanford, you have a better shot of getting in if you are also interested in tech entrepreneurship than a similar person who is not. Because of that, you should try to identify those programs where you match well.
As I mentioned, they expect to teach you how to do outstanding, publishable research in graduate school. They do not want to have to reteach what you learned during undergrad. Obviously everyone will have gaps and need to learn new topics, but if you didn’t learn what you studied in undergrad, it’s a red flag that you might need remedial instruction or be a poor self-taught learner. Neither of those bode well for graduate school, where you are expected to learn the necessary background while your advisor teaches you how to do publishable research. In effect, both of these are quick checks for basic competence, but compared to the rest of your application, these are relatively unimportant. A significant reason why incoming GPAs/GRE scores are so high at top schools is due to self-selection. People with a 3.8+ GPA think they have a shot at getting in, so they apply. I have heard several admissions folks say that they barely look at GPAs and GREs, especially at the top schools.
People spend an unnecessary amount of time worrying about the GREs. Keys to doing well: read widely and often for the verbal section, and review basic math and be careful on the math section. None of the math is complex or particularly challenging, but it is easy to make mistakes. Just practice a few times, be careful, and then focus most of your effort on the other parts of your application. It is quite possible to overcome a low GRE score, but it is not possible to overcome a bad statement of purpose, for example.
1) Understand what grad school will be like. Read a lot about it
2) Get research and work experience that closely matches the components of graduate school. Take initiative in everything that you do.
3) Make sure you present yourself as someone who will be great to work with (and actually be great to work with)
4) Demonstrate interest and know some of the open questions in the field
Note: I’ll probably continually update this with better/more complete advice.