First and foremost is the vertex, which is either the highest point or the lowest point of the parabola (which’s is what the graph of a polynomial looks like). From the vertex, you can always determine the next two points closest to the maximum/minimum.
Another key feature includes the y-intercepts, which are important if you want to find the zeros of the polynomial. Sometimes, the x-intercepts may come in handy as well, but not as often as the y-intercepts.