You got nothing wrong. To expand, you can add points on elliptic curves together. To add P and Q, you draw the line between them, find the third point on the line, and reflect it over the x-axis [1]. Then the idea is to find a generating set S of points so that every rational point on the elliptic curve is a sum of points from S. The size of S is called the rank of the elliptic curve.

[1] There are some details: https://en.wikipedia.org/wiki/Elliptic_curve#Group_law