Cartesian Product - Functions I'm comfortable determining whether a function is onto, one-to-one, neither or both with single variables such as $f(x)$. 
I'm a bit stumped when cartesian products and two variables are thrown into the mix. I don't quite get how something like the below would be determined.
\begin{align}
f\colon \mathbb{Z} \times \mathbb{Z} &\to \mathbb{Z} × \mathbb{Z}\\
 (x,y) &\mapsto (x+1,2y)
\end{align}
What would this translate to in plain English or a real world example?

In (more or less) plain English that is a function that takes as input a pair of integers and produces another pair as output. If you think of the variables as real numbers rather than just integers it maps the coordinate plane to itself. For example, the origin $(0,0)$ goes to the point $(1,0)$.
When thinking formally, every function is a function of a "single variable". In this case the variable happens to be a pair of numbers.
Don't be confused by the fact that the domain and codomain are ordered pairs with the way to describe a function as a set of ordered pairs. In this example the pair
$$
((0,0),(1,0)) \in (\mathbb{Z} \times \mathbb{Z}) \times ( \mathbb{Z} \times \mathbb{Z})
$$
is a member of $f$. But you will hardly ever think that way.
Whether this is a "real world" example is more a philosophical question than a mathematical one. I suppose you could think of it as describing how a frog hops from one point to another.
