We study the Riesz and logarithmic energies on the Grassmannian $\operatorname{Gr}_{2,4}$ of $2$-dimensional subspaces of $\mathbb{R}^4$. We prove that the continuous Riesz and logarithmic energies are uniquely minimized by the uniform measure, and we obtain asymptotic upper and lower bounds for the minimal discrete energies, with matching orders for the next-order terms. Additionally, we define a determinantal point process on $\operatorname{Gr}_{2,4}$ and compute the expected energy of the points coming from this random process, thereby obtaining explicit constants in the upper bounds for the Riesz and logarithmic energies.