I think none of the two answers so far really addresses the edited version of the question
which deals with infinite families of first order sentences.

The following graph properties (among others) can be expressed by infinitely many first order sentences, but not by finitely many (note that finitely many is equivalent to a single one):

(1) The graph is infinite: Take the collection of all sentences $\phi_n$ where $\phi_n$ states that the graph has at least $n$ vertices. A compactness argument shows that this is not finitely axiomatizable. (This is Kaveh's comment.)

(2) The graph is bipartite. Take the collection of sentences $\psi_n$ where $\psi_n$ says that the graph has no cycles of length $2n+1$. (This is Dave Marker's comment.)

It is possible to construct infinitely many different properties of graphs or directed graphs of this nature. Just take the property saying that for a fixed finite graph $H$,
the graph $G$ contains infinitely many disjoint copies of $H$.

It is worth pointing out that "the graph is finite" i.e., the opposite of (1), cannot be expressed by even infinitely many first order sentences.

(Given a first order theory $\Phi$ that is satisfied by all finite graphs, add infinitely many constants to your vocabulary. Then, by compactness, the theory consisting of $\Phi$ together with infinitely many sentences saying that the constants are pairwise distinct has a model, an infinite graph satisfying $\Phi$.)

Similarly, not being bipartite is not axiomatizable by a first order theory
(take an ultraproduct of the graphs $C_{2n+1}$, $n\in\mathbb N$, where $C_{2n+1}$ is the cycle of length $2n+1$).

Being connected is also not expressible even by an infinite first order theory.
I currently don't know whether non-connectedness is first order axiomatizable.
I asked this here.

is every first-order theory finitely axiomatizable?– François G. Dorais♦ Aug 28 '10 at 18:01