First remember that a space is first countable if it has a countable base locally, that is, for any point $x$, there is a countable collection of open sets such that any open set containing $x$ contains one of these open sets. Edit: These open sets are neighborhoods of $x$, not just any open sets. Thanks to Stefan H. for pointing this out.
A $G_\delta$ set is a set that is a countable intersection of open sets. Suppose you're given a single point $x$ in a first-countable $T_1$ space. What is a natural countable collection of open sets to try to intersect to get $\{x\}$? Why does this give you only $\{x\}$? (Hint: you haven't used the $T_1$ property yet).