You can prove essential smallness of the category of finite type $S$-schemes (without further assumptions), where $S$ is any scheme, as follows:

If $S$ is affine and we only consider affine finite type $S$-schemes, these correspond to finite type $\Gamma(S)$-algebras. These are isomorphic to algebras of the form $\Gamma(S)[x_1,\dotsc,x_n]/I$ for some natural number $n$ and some ideal $I$. These algebras form a set.

If $S$ is affine and we allow separated finite type $S$-schemes, these may be glued from affine finite type $S$-schemes $X_1,\dotsc,X_n$, $n \in \mathbb{N}$, and isomorphisms between the affine schemes $X_i \cap X_j \subseteq X_i$ and $X_j \cap X_i \subseteq X_j$. By the affine case, we can classify them by a set.

If $S$ is affine and we allow arbitrary finite type $S$-schemes, these may be glued from finitely many affine finite type $S$-schemes $X_1,\dotsc,X_n$, $n \in \mathbb{N}$, and isomorphisms between the separated schemes $X_i \cap X_j \subseteq X_i$ and $X_j \cap X_i \subseteq X_j$. By the first two cases, we can classify them by a set.

If $S$ is arbitrary, then finite type $S$-schemes may be glued from finite type $U$-schemes $X_U$, where $U$ runs through all open affines $U \subseteq S$, together with isomorphisms $X_U \times_U V \cong X_V$ for $V \subseteq U$. By the affine case, we may classify them by a set.

Remark that the category of locally finite type $S$-schemes is only essentially small when $S=\emptyset$: Consider $\coprod_{i \in I} S$ for sets $I$.

As Qiaochu points out, the category of (finite type) $S$-schemes is never small because it contains, for every set $X$, the scheme $(\emptyset,\mathcal{O})$ with the structure sheaf $\mathcal{O}(\emptyset)=(\{X\},+,\cdot)$.