Which infinity do integrals diverge to? When we say $\int{f}dx = \infty$, what is the cardinality of that $\infty$?

You are mixing up two notions of "infinity". One concerns the sizes of sets. A set is infinite (in that sense) when there's a one to one correspondence with a proper subset. That's the infinity that you mean when you talk about cardinality.
The other "infinity", written $\infty$, is sometimes confusing shorthand used when discussing limits. To say that a limit "is infinite" means that the quantity is eventually larger than any number specified in advance. $\infty$ is not a number of any kind.
Your confusion is quite common, and unsurprising. It's too bad we chose to use one word two different ways.
