The problem studied in this work is to determine the higher weight spectra of the Projective Reed-Muller codes associated to the Veronese $3$-fold $\mathcal V$ in $PG(9,q)$, which is the image of the quadratic Veronese embedding of $PG(3,q)$ in $PG(9,q)$. We reduce the problem to the following combinatorial problem in finite geometry: For each subset $S$ of $\mathcal V$, determine the dimension of the linear subspace of $PG(9,q)$ generated by $S$. We develop a systematic method to solve the latter problem. We implement the method for $q=3$, and use it to obtain the higher weight spectra of the associated code. The case of a general finite field $\mathbb F_q$ will be treated in a future work.