To write a high scoring ACT essay, time is of the essence as is practice preparation. Many students panic when they realize how short the time is to think, outline ideas, write and proofread the essay. My first point would be to not panic. After reading the question very carefully so that you know exactly what the question is asking, take a deep breath and calm your mind. The minute you spend doing this pays off in the end. Then, write notes to see if you can support the thesis you have in mind as a response. If you have the three ideas you need, try to begin your essay in an unususal way such as a comparison with your life or an important lesson learned which fits this essay. As you compose, don't let yourself forget to watch the clock as you must not leave a section of this essay unfinished. Keep each section of the essay sort of on a timer with so much time for the body paragraphs, so much time to make the introduction and conclusion match up, and so much time to proofread. Don't spend time with perfect handwriting but do make it readable. Do spend a few minutes proofreading so that the people who read the essay understand that you know how to make your essay better. Think about complex sentences, word choice and whether your essay truly answers the question asked.
The two previous answers to your question are full of excellent advice. I would like to add one observation based on my personal experience. I believe there is nothing more helpful in becoming a competent writer than doing lots and lots of writing. A serious student should form the habit of writing a certain quota of words every day, just so that the act of writing does not seem strange and weird, like walking a tightrope across the Grand Canyon, when the time comes to write that essay exam, or take-home exam, or whatever. The word-quota should be long enough but not too long. Regularity is more important than the old tortoise-and-the-hare routine. I guess three or four hundred words per day would be a reasonable quota. The easiest way to meet such a quota would be to keep a diary or journal. I'm sure that many English teachers have made this suggestion many times. I don't think there is any substitute for practice. A good model to follow would be Anne Frank's Diary of a Young Girl. She writes about anything she wants to. Another good model is Somerset Maugham's A Writer's Notebook. Keeping a journal or diary is not only a good way of becoming a competent writer, it is a good way of getting to know yourself. It might seem difficult at first, but it becomes easier, and nearly addicting, with time.
Because a student has only 30 minutes in which to write an ACT essay, it's important to be very focused and think ahead. The five paragraph format works quite well for this.
First, read over the question carefully. Think of a simple, clear solution that you can support with two or three examples. Then write an outline in the form of a heading for each paragraph. Compose the essay by writing 3-4 sentences after each heading. If you get stuck on one heading, write the .next section, and then move back. Don't overwrite one point and run out of time.
Finally, take time to proofread.
A good idea is to practice this sort of timed writing with an egg timer at home to get comfortable with it before you take the test.
As aforementioned earlier, essays should be practiced on a regular basis. It's good to be accustomed to writing in general. Writing ACT essays will get you ready for, not only the types of questions that will be on the test, but also the average amount of time it will take for you to finish it. I highly recommend, however, that you not just write regularly, but also read out of habit, too. It does something akin to sharpening the mind and will help you, especially those who struggle with the english language. Please be concise and to-the-point. When making an argument, have hard proof (with quotations, if possible). Do not wrestle with one part of the essay, and move on. When you finish, go back and edit. Before attempting an essay, read the rubric.
To properly write an effective, high-scoring ACT essay, you need to hit several key points. Those points are: conciseness, cohesiveness, and clearness. That is, the overall essay must be detailed, yet not have unneeded, rambling phrases or sentences. Also, the essay has a whole has to make sense and flow in a reasonable, smooth manner. The graders on the ACT look for the overall sense of the essay, so don't worry too much about grammar and proofreading. You should however use effective syntax. That is, you should try to vary your sentence lengths, and use varying forms of diction throughout your essay. All-in-all, to obtain a high-scoring ACT essay, just write an essay that answers the questions that has plenty of concrete details and stays at a narrow scope of the subject.