Note: since I was a physics major, and since the only courses I've taught have been in physics, the ideas presented below make the most sense when applied to physics, enginnering, and math courses. Some of them are general enough to apply to others, but I think my observations are really best suited to what I have the most experience in.
No, I don't mean "the right way" and "the wrong way."
When I was teaching, I had a couple of students each semester who came to me almost every week for help. It took a lot of time, but it was very valuable experience: it gave me a much better idea of just where students were getting confused and how to fix it. It was mostly through them that this and all of the other tips came.
Some people are good at looking at a set of symbols, picking up the rules as to how they fit together, and working out the problem from there. I and most of my students, however, prefer to have some mental picture of a process to go through. When trying to explain how to do a problem, I would sometimes get blank looks, or a response like "I'm just no good at this sort of stuff." However, I was always able to keep explaining, each time taking a different approach. Eventually, either I was able to figure out just where the block was and how to explain it away, or else they figured it out. Sometimes, I couldn't be sure if they really got it, so I would present another problem which focused more specifically on where I thought the problem was. Once they did "get it," they could proceed through the problem, writing down almost exactly the same set of equations they didn't understand to begin with, but this time, they could make a logical argument on their own. Presenting things in a variety of ways has never failed me yet in trying to get a point across, so if anyone tells you, "this is the way to do the problem," well, they may be right in terms of what you would see on paper. But the mental process to get there could take any number of different forms.
This process has been extremely useful to me in almost all of the courses I've taken, and certainly in every course I've taught. Writing down what you're trying to find at the top of the page can keep you focused on your goal, rather than spending huge amounts of time going in circles, never really coming to an answer. And listing what you're given can help you figure out just what kind of problem it is you're trying to solve, and what equations to use.
This was the one I had the hardest time convincing my students of. This is an engineering school, and many engineering courses don't seem to worry much about units (at least that's what my students told me). All too often, if someone would come up with the right number, they'd think it was right. But when I ask if it's in millimeters, centimeters, meters, or kilometers, they would often have no idea. Every one of my quizzes had stated on it "use units in every step." I wasn't too harsh with that unless there weren't units in the answer, but there have been times when students have been wrong by factors of a million because of carelessness about units. Many students have said that they don't use units because it confuses them: they would get the right answer without using units; but when they do use units, they think the answer is wrong. I've actually seen this happen. But much more often, using units in every step can tell you whether or not you've made a stupid algebra mistake (this has saved me more times than I can count), and if you did, just where it happened. Here's an example of how I'd try to teach about units:
Say you knew something was moving at 5 m/s (meters per second), and that it moved a distance of 10 inches. How much time did it take to move those ten inches?
First, you need to know that v=x/t (velocity = distance travelled / time spent travelling that distance). Multiply both sides by t, divide both sides by v:
t=x/v: this gives: t=(10 in)/(5m/s) = 2 (in·s)/m. This is true, but useless. Nobody measures time in second-inches per meter. So you must convert that to seconds. To do that, you may remeber that there are 2.54 centimeters per inch, and that there are 100 centimeters per meter.
Most people easily accept that if you know the equation:
x=y
is true, then
x/y = 1
is also true. In the same way, we know that:
2.54 cm = 1 in., so:
(2.54 cm)/(1 in.) = 1
Multiplying by the ratio (2.54 cm)/(1 in.) doesn't change the content of the number you started with, only the way it's represented. It's exactly the same as multiplying by 1. A lot of students really had trouble with this idea, but I think it's a valuable one.
So to get the time, we take:
t = 2 (in.·s)/m · (2.54 cm)/(1 in.) · (1 m)/(100 cm)
Or, take our previous answer, and multiply by 1 twice (the conversion factors). Cancel out the inches, the centimeters, and the meters, giving us:
t = 0.0508 s
This is very different from 2 s. If our future engineers don't learn this lesson, I worry about many of the things being built.
(Note: there are exceptions to this rule, but not until you hit graduate level physics; and even then, it's almost always true.)
On almost every quiz and exam I ever gave, all of the equations which I thought might be needed were given. My first semester teaching, I only gave a small set of equations. In later semesters, I gave them every significant equation we'd ever had up to that point. And I ended up getting some pretty bizarre results. For example, W is often used to stand for work (closely related to energy), weight (a force), and watts (a unit of power). I've seen these things exchanged in the most bizarre ways. And "m" might stand for meters or mass, but never for momentum (for which "p" is used, since the days of Newton). Looking at the units of the parts of the equation can help there if you do forget, but usually, try to at least remember the general subject of the chapter that each equation came from.
This may sound typical or obvious, but there's a reason beyond just getting the grade on the homework and class participation. Even if there is no graded homework, it's useful. Why? Well, many professors and graduate students may not be the most creative people in the world. Or else there may be time pressures. Or, a lot of effort might have gone into choosing good problems to assign for homework. But for whatever reasons, it's often the case that quizzes and exams are not-very-significant variations on the homework problems. This is going to vary greatly from teacher to teacher and course to course, but it's true often enough to be worth thinking about. There are certain key points that need to be taught in each course, and to make sure they're learned, they're often tested in a variety of ways. This is generally a good thing for both teachers and students. And being aware of it in those courses where it's true can help give you that extra bit of motivation to actually learn from the homework assignments.