Algebraic geometers are often interested in certain “positive” cones that one can associate to varieties, such as the nef and effective cones. While traditionally studied in the divisor and curves cases, cones of cycles of intermediate dimension have recently been the subject of much study. We discuss some results on nef and effective cones of intermediate dimension cycles in the case of Hilbert schemes of points. In particular, we look at the Hilbert scheme of 3 points in $\mathbb{P}^3$. Our computation generalizes results of Ryan & Stathis and requires a careful analysis of the PGL orbits in the Hilbert scheme, as well as a new basis of the Chow ring inspired by Mallavibarrena and Sols.