$\DeclareMathOperator\Spec{Spec}$In the case of an affine group scheme $G$ acting on an affine scheme $X$ over an algebraically closed field $K$ you can ask what the $K$-points $\Spec K \to X//G$ of the stacky quotient are. By definition they consist of pairs of a $G$-torsor $P \to \Spec K$ over $\Spec K$ and a $G$-equivariant map $P \to X$. Under mild hypotheses on $G$ (I think it suffices that $\mathcal{O}(G)$ be countably generated, if I'm reading <a href="https://arxiv.org/abs/1304.6216">Deninger - A remark on the structure of torsors under an affine group scheme</a> correctly), $G$ is the only $G$-torsor over $\Spec K$, so the $K$-points of $X//G$ are $G$-equivariant maps $G \to X$. 

These correspond exactly to the $K$-points $X(K)$ of $X$, together with the action of $G(K)$ on them (acting on $G$ from the right); in other words, $(X//G)(K)$ is the stacky quotient / action groupoid / homotopy quotient $X(K)//G(K)$, and in particular its $\pi_0$ (set of isomorphism classes) is exactly the set of orbits of $G(K)$ acting on $X(K)$.

(Actually I don't know if I need to assume $X$ affine here but I'm playing it safe.)