Let $G$ be a group acting on a ringed topological space $(X,\mathcal{O}_X)$ and let $p: X \mapsto Y=X/G$ endowed with the quotient topology. Clearly, the action is $x g=g^{-1}(x)$ It's also clear that $p$ is an open map and that $G$ acts both on $p^{-1}(V)$ and $\mathcal{O}_X(p^{-1}(V))$.
1) Let $V \subset Y$ open and set $\mathcal{O}_Y (V)=\mathcal{O}_X(p^{-1}(V))^G$. I have to show that $\mathcal{O}_Y$ is a sheaf. In order to do this, I consider a section $s_{V \cap V_i}=0$ for every $V_i$, open subsets of a covering of $V$. By our definition, we have (?? I'm not sure) $s_{p^{-1}(V) \cap p^{-1}(V_i)}=0$ which is $G-$invariant. But $\mathcal{O}_X^G$ (I use this notation to denote the sheaf $\mathcal{O}_X(p^{-1}(V))^G$, when $V$ varies) is a subsheaf of the sheaf $\mathcal{O}_X$, so I obtain the thesis. But I'm not convinced of my proof! (I have only proved the uniqueness of global section satisfying sheaf axiom, its existence is proved in the same way)
2) After that, it's easy to show that $(Y,\mathcal{O}_Y)$ is a ringed topological space. I have to prove that $p$ induces a morphism of ringed space $(X,\mathcal{O}_X) \mapsto (Y,\mathcal{O}_Y)$. I have to prove that the stalk map $p^*: \mathcal{O}_{Y,p(x)} \mapsto \mathcal{O}_{X,x}$ is a local homomorphism, but I can't do it.
3) $p: X \mapsto Y$ verifies the universal property of the quotient. If I were working with affine schemes, the result would be obvious thanks to the the contravariant functor between the category of affine schemes and that of rings and using the universal property of rings, according to which any morphism $\phi: R \mapsto A$ such that for all $g \in G$ we have $g \phi=\phi$ factors uniquely through $A^G$. But in the general case of ringed spaces, I can't find a so simple proof.