There is a smart way of learning, and it consists in first finding out what are the most valuable pieces of knowledge to acquire. The ones that will give you the highest value for your investment in attention and time. For instance, when you are learning a new language, it is typical to find that 2% of the words make 95% of conversations. You'd do far better if you knew what those words are when acquiring vocabulary.
My question is whether there are mathematical concepts, results, techniques, tricks, etc., that give such high payoffs. It could be because they are used everywhere, or because they are really easy to learn considering how much they worth, etc. As was pointed out in the comments, the meaning of "payoff" depends on the goals of your mathematical tools. Techniques like frequency analysis might bring the bread to the table of an engineer, but be irrelevant to a number theorist. I, in particular, plan to become a sort of applied mathematician in neuroscience and perhaps other "soft" sciences. I don't know beforehand exactly what I will be doing. That's why I want to gather a set of skills that are useful in a broad, applied, sense.
You might express the efficiency of a bit of knowledge in the following way:
$$\frac{\text{how much is it worth mathematically}}{\text{how much do I need to invest to learn it}}$$
The "how much is it worth" would be the weighted sum of
- The extension of mathematics in which is required to know it
- The number of applied problems whose solutions require it
- The work time you could save by using it, instead of acting in a naive way
- The average amount of errors you could stop making if you use it
The "how much do I need to invest" is the time it would take me to master this skill, trick, knowledge, etc. provided I already have all the prerequisites assumed in a standard introduction to the topic. Avoid considerations about how much money books or papers would cost, that is not the issue here.
Current List (to be filled with the answers)
- Visual evaluation of 2D and 3D graphs (see Jasper's answer)
- Dimensional analyis: you will have an edge in problems related to physical magnitudes, with little extra effort. You could also find the functional form of some solutions without calculations
- Logic: most common and dangerous fallacies
- Algebra: Elementary Algebra (how to manipulate expressions, and doing it without errors) Linear Algebra (Matrices, Vector Spaces, etc.)
- Calculus: The very vital concepts of limit, derivative and integral. Numerical procedures
- Statistics: How to lie, Resampling techniques(from Richard's answer)
- Optimization:
- Approximation methods: like in Mahajan. Street fighting mathematics, they are easy to use, but can give you a decent answer to a hard problem
- Fourier Transforms: are used in a lot of areas and not hard to grasp
- Elementary Programming: It's not properly maths, but needed everywhere. You could learn high level open source languages like Python and R in around a semester, and use them for a lot of things.