By definition, the cardinality of set $Y$ is smaller than the cardinality of set $X$ exactly if there exists an injection from $Y$ into $X$. Obviously the image of $Y$ in $X$ then has the same cardinality as $Y$. Thus yes, if $k<|X|$, then $X$ has a subset of cardinality $k$.
And this indeed means that $\mathbb R$ has a subset of cardinality $\aleph_1$. But we cannot explicitly give one. Or more exactly, we cannot give an explicit set of which we can prove that its cardinality is $\aleph_1$. Of course since it is consistent that $\mathbb R$ has cardinality $\aleph_1$, just $\mathbb R$ itself might be such an example. But we cannot prove or disprove it.
Note that my first paragraph is independent of the axiom of choice. However, the second paragraph isn't, as without the axiom of choice it is consistent that $|\mathbb R|\ngeq\aleph_1$.