Given your mentioning of the word "countable" (and looking at the link), I think that the "differing types of infinity" that you are thinking of are different infinite cardinalities (i.e. different possible infinite sizes of sets). This notion of infinity is not the same as the notion of infinity that is used when one says that a limit approaches infinity. (Of course, they are related at a basic level, in that both allude to the idea of being larger than any finite number, but at a more technical level, they are different notions.)
Ronaldo in his answer has given an explanation of what it means for
$\displaystyle \lim_{t\to\infty} f(t)$ to equal $\infty$, and this is just as valid for $e^{t^2}$ as it is for $e^t$.
As Dan Ramras notes in one of his comments, the rate at which $e^{t^2}$ approaches infinity
is much greater than the rate at which $e^t$ approaches infinty (which is turn much
greater than the rate at which $t$ approaches infinity, for example). In terms of Ronaldo's answer, this means that for a given $M$, the $\epsilon$ that you have to choose
to be sure that $e^{t^2} > M$ when $t > \epsilon$ is smaller than in the corresponding situation for $e^t$. But both quantities approach infinity as $t$ does.