I am assuming the first equation is the parabola y = x2 - 3. To graph a parabola I always recommend starting with the vertex and then plotting at least 2 other points.
One way to find the vertex is to put your equation in the form y = (x - k)2 + h where the vertex is the point (k,h). We see that this can done easily by simply by saying y = (x - 0)2 - 3. This gives us that the vertex is at the point (0,-3). Now that the vertex is known choose two other x values (one on each side of the vertex) plot those points by plugging them into the equation and that should be enough to get a rough sketch of the parabola.
Next we must graph the linear equation y = 2x. For any linear equation we always start with y - intercept and from the there since the slope is 2, we go up 2 and over to the right 1 to get the next point (repeat again for a 3rd point if you would like).
Finally once both graphs are sketched, the solutions to the equation x2 - 3 = 2x will be the x values where they intersect.