Which infinity is meant in limits?

There is no need to ask about countable or uncountable infinity. The symbol $\infty$ here is used with the following precise meaning.

Let $f$ be defined on some ray $(a,\infty)$. We say that $$\lim_{x\to\infty}f(x)=A$$ if there exists $A\in\Bbb R$ such that for each $\epsilon >0$ there exists an $M>0$ for which $$x>M\implies |f(x)-A|<\epsilon$$

That is, for any $\epsilon >0$ we're given, we can take $x$ large enough to make $f$ as close as we wish to $A$.

The more "dramatic" (symbolically) $$\lim_{x\to\infty}f(x)=\infty$$ means precisely that

... for each $N>0$ there exists $M>0$ such that $x>M\implies f(x)>N$.

That is, we can make $f(x)$ as large as we want by taking $x$ large enough.

ADD Compare the above to the notation $$(a,\infty)$$ I used in it. We're not wondering about any "infinity" but just about the set of numbers $x>a$. The symbol $\infty$ is convenient and intuitive. Brian uses $(a,\to)$ instead!

The $\infty$ in that limit does not refer to an infinite cardinality at all. The expression

$$\lim_{x\to\infty}f(x)=L$$

is simply an abbreviation for the following statement:

$\qquad\qquad$for each $\epsilon>0$ there is an $x_\epsilon\in\Bbb R$ such that $|f(x)-L|<\epsilon$ whenever $x\ge x_\epsilon$.

As you can see, there is no infinite anywhere in that statement. The ‘$x\to\infty$’ in the limit notation is a reminder that we’re talking about what happens when $x$ is very large; it does not refer to a specific entity $\infty$.

(There is in fact a way to interpret this $\infty$ as a specific entity: one can replace $\Bbb R$ with the so-called extended reals, which include two new points $\infty$ and $-\infty$. In effect this adds an endpoint at each end of the real line. But these objects, despite their standard names $\pm\infty$, are simply points in an extended space, not cardinal numbers that can be used for counting infinite collections.)

None of the above. That's why it neither says $\omega$,nor $\aleph_0$, nor $\mathbf c$. Rather, $\infty$ should be considered as a symbol. There's no infinity really used in the definition: $$\lim_{x\to\infty}f(x)=c\iff\forall\epsilon>0\colon\exists M\in\mathbb R\colon\forall x>M\colon |f(x)-c|<\epsilon.$$

The infinity here is a point at infinity in the two-point compactification $\mathbb{R} \cup \{ -\infty, +\infty \}$ of the real line. This is just one of many meanings of "infinity" in mathematics; see, for example, this math.SE question.