Open Sets in Product Topology.
As Hui Yu says, you are not going too far with just the definition and set-theoretic operations: think about specific examples.
For instance, before fighting against scary monsters like infinite arbitrary products, how about looking for examples in the humble $\mathbb{R}^2$? Are you sure you could find out a simple characterization of the open sets there (which happen to be the same for the product topology and for the Euclidian, usual one)?
E.g., what about a set like this one:
$$ U = \left\{ (x,y) \in \mathbb{R}^2 \ \vert \ xy > 1 \ , \ x > 0 \right\} \quad \text{?} $$
It's an open set. Do you think you could describe it easily (I mean, without just repeating the definition of open sets in $\mathbb{R}^2$) in terms of the open sets of the basis of the product topology?
Note that the following are true (in all spaces): (fix a base $\mathcal{B}$ for a space $X$)
$f: X \to Y$ is open iff $f[B]$ is open in $Y$ for every $B \in \mathcal{B}$.
$f: X \to Y$ is continuous at $x$ iff for every open set $O$ that contains $f(x)$, there exists some $B \in \mathcal{B}$ such that $x \in B$ and $f[B] \subset O$.
$X$ is compact iff every cover of $X$ with elements from $\mathcal{B}$ has a finite subcover.
$D \subset X$ is dense iff every non-empty $B \in \mathcal{B}$ intersects $D$.
$f : Y \to X$ is continuous iff $f^{-1}[B]$ is open in $Y$ for all $B \in \mathcal{B}$.
Note that we can reason about continuity, openness, compactness, knowing only a base for the topology. So in most cases, all we really need is a good description for a base.
This is analogous to the situation of metric spaces $(X,d)$, where a base is specified (all sets of the form $B(x,r) = \{ y \in X: d(x,y) < r \}$, where $r>0$) and continuity between metric spaces is often expressed using the $\epsilon$-$\delta$ definition, which is just like 2., except using this base in both spaces. For product spaces as well, all proofs involving them essentially uses this base (or the subbase of all $\pi_s^{-1}[O]$ for open sets $O$ in $X_s$). We really do not need a description beyond the fact that they are unions of basic sets.