The computation of explicit equations of secant varieties is hard in general.
However, things are simpler when one works with *determinantal varieties*, i.e. varieties whose equations are given by some minors of a matrix of homogeneous forms.

In fact, there is the following result, that one can find in [Harris, Algebraic Geometry, p. 145]:

**Proposition.** Let $M$ be the projective space of $m \times n$ matrices and let $M_k \subset M$ be the subvariety of matrices of rank at most $k$. Assume that $2k < \min\{m, n\}$. Then the secant variety $S(M_k)$ is equal to the subariety $M_{2k} \subset M$ of matrices of rank at most $2k$.

As an example, let us consider the Veronese surface $X \subset \mathbb{P}^5$, whose equations are the $2 \times 2$ minors of the matrix
$$ M:=\pmatrix{z_0 & z_3 & z_4 \cr z_3 & z_1 & z_5 \cr z_4 & z_5 & z_2}.$$
In this case $m=n=3$ and $k=1$, that is $X=M_1$. Then the secant variety $S(X)$ coincides with the determinantal variety $M_2$, i.e. with the cubic hypersurface defined by $$\det M=0.$$
This is essentially due to the fact that the linear combination of two rank $1$ matrices can have rank at most $2$.

Developing the determinant along any line or column, you can find an expression of the equation of $S(X)$ as a linear combination of generators of the homogeneous ideal $I(X)$.

Moreover, in the same way one can find equations for the secant varieties of rational normal curves, see [Harris, Algebraic geometry, p. 103].

For a deeper treatment of this problem, see the paper by Ottaviani and Landsberg *Equations for secant varieties of Veronese and other varieties*, arXiv:1111.4567.