I think your last paragraph was most of a good answer. i.e. a graph should contain the points of interest.
Linear Equation - X and Y intercepts, and depending on the lesson topic, highlight points that were part of the problem as stated. (e.g. the problem statement may have been "given the 2 points, solve for the equation of the line and produce a graph. In this case, you can ask they highlight those 2 points on the line.)
Parabola - Typical to ask for "five points" - Vertex, 2 X-intercepts, Y-intercept, and symmetric point. [Note, this is the generic approach, of course if the parabola doesn't intersect X-axis, those 2 points don't exist.] Focus/directrix for the higher level.
Cubic - The X-intercepts, and Y-intercept if appropriate. Let me offer a graph of a potential problematic situation.
I say "problematic" for one common issue I see. A teacher might give a worksheet/exam where the blank graph has a scale already printed. That scale can be 10X10 or 20X20 and it would cover the 3 X-intercepts, but not let the student draw the local extrema. The request sometimes feels ambiguous to the student. A 'sketch' might mean general shape of the curve and identify X-intercepts, along with Y (with a note on the graph, indicating "-40"). For advanced students, usually pre-calc, the exact extrema values are important as well. And note that to show them in a pleasant graph, the scales are different. I'd prefer graphs with unlabeled axis for the more advanced students' use.
For the other conics, circle/ellipse, hyperbola, a similar approach. These have their own points of interest, foci, vertex, directrix, asymptotes, and it's important to have problems be clear, i.e. if they are given the graph, those points should be labeled or identified, if they are given the equation, their graph should should these points and the drawing itself be neat.
Trig graphs - There are a number of directions problem come from. "Graph this equation", "Look at this graph and write the appropriate equation", "Here is a word problem. Graph the sinusoidal function, and give the equation."
In each case, I'd suggest (a) Identify the minimum and (b) maximum values, as well as the (c) sinusoidal axis / midline. Next, Identify the (d) period. From there, the appropriate starting point (the horizontal shift) is determined and the graph drawn. For some problems, one period is enough, others, multiple periods. I'd note that if min/max if a high enough number, the graph need not always show the X-axis, only the scale for x values. e.g. a value ranges from 100 to 110 over time. I've seen frustrated students produce a scale that has a blank area from 0-100, and a tiny graph above that. Better just to show them that not all graphs require starting at zero at either axis.