I teach high-school calculus, and many questions can be checked by a graphing calculator. So my first strategy is to teach the use of the calculator and using it to check answers when possible.
However, often a student will see that his final answer disagrees with the calculator but he does not know which of his steps introduced an error.
I have found that I need two more strategies to avoid this. First, I encourage my students to check some of their steps if the final answer seems wrong. By using a "binary search" they can narrow the problem to just one step and sometimes even to a part of one step. They can then focus their attention to figure out the exact error.
Let me explain the search for errors more. The best steps to check are the key ones. (For example, in finding a normal line to a point given by its $x$ coordinate on the graph of a given function, the steps to check are the $y$ coordinate of the point, the derivative of the function, the slope of the graph at the point, the slope of the perpendicular line, and finally the equation of the normal line.) If there are no obvious key steps, the student should check a step near the middle of his calculations. If that step checks, ignore the steps in the first half and check the step halfway into the second half; if it does not check, ignore the second half and check the step halfway into the first half. At each stage decide which half of the remaining steps hold the error and cut that group into half. (For example, if there are 16 steps then one possible series of checks is 8 which doesn't check, 4 which does, 6 which doesn't, and 5 which doesn't. There then must be an error in step 5.) This halving strategy is similar to the "binary search" used in computer science. When a step is pinpointed, sometimes it can be broken down into smaller steps which can be checked.
Another strategy is to teach a mild distrust of the calculator. Sometimes the answer is correct and the calculator is wrong. I teach them several reasons the calculator can be wrong. The student may have mistyped the expression (the most common problem). The student may misunderstand the proper syntax ($xsin(x)$ is not interpreted as $x\cdot \sin(x)$ but as a new function $xsin$). And occasionally the grapher simply makes its own errors (smoothing out a discontinuous function, not extending a graph close enough to a singularity, etc.).
So, overall, I try to teach using the graphing calculator as a helpful but not entirely reliable tool whose main purpose is to check answers gotten in another way. I teach these concepts by doing them in practice problems at the board. I let students guide the solution, and sometimes they get it wrong. I often just proceed, and when we discover that we have gone wrong somewhere I demonstrate how to find the error. I deliberately choose some examples that lead to grapher error, often examples that came up in previous years. This way I do not even need to be embarrassed when I make a mistake: it leads to finding the bug, and I give extra credit to the person who discovered that there was a mistake.
Details: I use the TI-Nspire CX calculator, without a computer algebra system. So far this seems to have the best mix of power in graphing and in numerical calculations without allowing the student to avoid doing his own algebra. At least, it seems the best mix for high school precalculus and calculus. I use the Teacher Software to emulate the calculator on my school computer and project the results on the whiteboard. It took me years to get the funds for all this but I have a pretty decent setup now.