Let $G$ be a group that acts on $X$. I want to show that the orbits of $G$ partition $X$. I am given the relation $x\sim y \iff x\in Orb(y)$. Now:
$x\sim y\iff x\in Orb(y) \iff x=gy$ for some $g\in G \iff g^{-1}x=y$ for some $g\in G$ $\iff y\in Orb(x) \iff Orb(x)=Orb(y)$.
So I use this to show the three conditions of an equivalence relation:
$x\sim x \iff Orb(x)=Orb(x)$
$x\sim y \iff Orb(x)=Orb(y) \iff Orb(y)=Orb(x) \iff y\sim x$
$x\sim y, y\sim z \Rightarrow Orb(x)=Orb(y)=Orb(z) \Rightarrow Orb(x)=Orb(z)\Rightarrow x\sim z$
Does this look okay?