As you should, you prove the Nullstellansatz Nullstellensatz early on, as the statement that the closed points of $\mathbb{A}^n_k$ are in bijection with $k^n$, for $k$ an algebraically closed field. I wonder whether it is also a good idea to say that, for any $k$, the closed points of $\mathbb{A}^n_k$ are in bijection with the Galois orbits in $\overline{k}^n$. This might require too big a digression into Galois theory, but I remember a number of my grad school classmates having confusions about closed points over non-algebraically closed fields which could be immediately answered from this decsription.
As you should, you prove the Nullstellansatz early on, as the statement that the closed points of $\mathbb{A}^n_k$ are in bijection with $k^n$, for $k$ an algebraically closed field. I wonder whether it is also a good idea to say that, for any $k$, the closed points of $\mathbb{A}^n_k$ are in bijection with the Galois orbits in $\overline{k}^n$. This might require too big a digression into Galois theory, but I remember a number of my grad school classmates having confusions about closed points over non-algebraically closed fields which could be immediately answered from this decsription.