You have been given a lot of great tips from the above posters. I would add that you shouldn't be afraid to ask a teacher to slow down or repeat something if you think it is important and should be written down. Sometimes, teachers get caught up in what they are saying and forget that students need us to slow down so we can write, but mosts of us have no problem with you speaking up and asking us to slow down.
Learning how to paraphrase- especially in an English or History class- might be your best tool. If a teacher has a long paragraph on a projector screen, listen to what he/she says about that paragrapha and write the general idea of what the paragraph says in your own words. Never feel like you have to write everything down word-for-word in ANY note-taking situation. If you do, you will be so focused on getting the words down that you won't have heard a word the teacher said, and you probably won't be able to read your own notes. PARAPHRASE! Also, use abbreviations that you can recognize so you don't have to write down full words.
I also highly recommend the two-column note (or Cornell) note-taking system. You've already had a couple of other teachers recommend it, but from a former student perspective, I used it in college and high school and it genuinely helped me organize my thoughts. Now that I'm a teacher, I sometimes go back to look at old notes from a class, and I can still see what the main ideas are.
The key is figuring out "what the important things" are, which is easier said than done but can become easier with practice.
If you have the opportunity to read assigned material before you come to class, that should give you a basis of knowledge that will help you sort out the important information in the lecture from the not-as-vital. As mentioned in #6, if the lecture is accompanied by a PowerPoint or other visual outline of the lecture, that's a great way to see what the instructor considers the critical points - which would make them good for you to have in your notes!
I can also endorse the Cornell, or dialectic note-taking system. I always used it, especially when I taught high-school freshmen. Generally speaking, the best way to know what is important is to listen for verbal cues like some of the other responses have mentioned. Obviously, things that are written on the board (or, these days, on Powerpoint) are important too. Another hint would be to complete assigned readings ahead of time. Often lectures will refer back to them, and they can provide context for what is important.
This really is an awesome question, and kind of a tough one to answer. We've all had that situation where we look back upon our own notes and get the feeling that we might have missed something important.
I will definitely second many of the important tips that speamerfan brought up. Making preliminary outlines based on the course reading, copying down any board notes, and listening for those repeated or emphasized things will really help you figure out what's important. I know that when I'm teaching, in a lecture scenario, I actually say, "Now this is really important... [insert idea here]." If you really listen, I bet your teacher or professor will probably give a similar "hint" around significant points.
One more thing you can investigate, which always worked great for me, is the Cornell Notetaking System. Basically, this involves drawing a line down the middle of your page so that your notebook page is divided into a left side and a right side. On the left side, you can write down key words, phrases, topics, or questions, leaving several lines between each entry. On the right side, you fill in more information about each word, topic or question--and this can be done either during the class or afterword... This is helpful if you have a very speedy talker for an instructor. Below is a link to more info about Cornell Notes. Check out the links under the heading "External Links" for some templates so that you can see if Cornell is a good method for your notetaking needs!
The best thing to do in this instance is to make sure that you are clear on what the topic is. Usually, a teacher or professor will give you something (whether it be on paper or verbally) that lets you know what you will be learning about. If you are clear on your learning goals, you can usually figure out what's important to write down.
Also, figure out a way to write notes briefly, a kind of personal shorthand.
This is a great question! It is impossible to write down every word your teacher says, and it isn't necessary, either. But there are some strategies you can use to get down what is important, no matter what the class is. There are also good strategies for specific kinds of classes.
First, if you read in advance what the class is going to cover and create an informal outline in your notebook, you know in advance what is important in the reading materials, and when the teacher discusses those topics, all you have to do is fill in the notes in those sections.
Second, whatever the teacher writes down on the board is important and should be written in your notes.
Third, teachers have a way of repeating important points or raising their voices a bit when they come to important points. Once you get to know a teacher a little better, you will see that there are always these little clues.
Fourth, some teachers provide handouts that allow you to see what is important during their lectures. However, you should still take notes on the areas covered by the handout because this is another way knowledge gets into your brain, through your fingertips.
In any course in which there are definitions you must learn, always take notes on definitions. That is the vocabulary of the class, and you cannot learn without understanding that vocabulary.
In math classes, the examples written on the board should always be copied into your notes, along with what the teacher has to say about doing the example problem. You may think you understand the problem and how to solve it, but sometimes this knowledge slips away later on, when you have to do that kind of problem yourself. Formulas and what the teacher has to say about them should be part of your notes, too.
Good note-taking is a valuable skill in high school, in college, and even at work. It's wonderful that you are asking about this now, so you can begin to hone your skill.