I have coded an implementation of elliptic curves in order to apply some of the ECC algorithms. However, in most of them Alice needs to choose a point P on a given curve. What is the general procedure for selecting such a point?

Given a small example such as y^2 = x^3 + x + 1 over $F_{25}$, is there an algorithm to generate a random point on the curve? In my implementation points on this field are represented by polynomials, if that is relevant.