In order to show that $\beta(\mathbb Z)$ is not first countable, it will suffice to show that $|\beta(\mathbb Z)|\gt2^{\aleph_0},$ since a Hausdorff space which is separable and first countable has cardinality at most $2^{\aleph_0}$ (each point is the limit of a convergent sequence of points in a countable dense set).
The space $C=\{0,1\}^\mathbb R$ is a separable compact Hausdorff space. Define a countable dense subset $S\subseteq C$ and a surjection $f:\mathbb Z\to S.$ Since $\mathbb Z$ is discrete, $f$ is continuous, and therefore extends to a continuous surjection $g:\beta(\mathbb Z)\to C,$ showing that $|\beta(\mathbb Z)|\ge|C|=2^{2^{\aleph_0}}\gt2^{\aleph_0}.$
Since you already know about the Alexandroff one-point compactification, let me begin by saying that the Stone-Cech compactification is at the other extreme, adding as many points at infinity as possible. To see what that could mean, let's consider some other compactifications of $\mathbb R$, starting with the most familiar, the extended real line, obtained by adjoining the two points $+\infty$ and $-\infty$ at the two ends of the line. Compared with the Alexandroff compactification, we're now distinguishing two different ways of "going to infinity". Some sequences that converged to $\infty$ in the Alexandroff compactification fail to converge in the extended real line because part of the sequence goes to the left and part to the right (e.g., $(-1)^nn$).
This idea can be extended, to produce "bigger" compactifications. If you visualize $\mathbb R$ as embedded in the plane as the graph of the sine function and then take its closure in the extended plane $(\mathbb R\cup\{+\infty,-\infty\})^2$, you get a compactification with a whole line segment at $+\infty$ (and another at $-\infty$). Similarly, embedding $\mathbb R$ in $3$-dimensional space as a helix, by $x\mapsto (x,\cos x,\sin x)$, we get a compactification with circles at the ends. Proceeding analogously with all bounded continuous functions $\mathbb R\to\mathbb R$ (in place of $\cos$ and $\sin$), in a very high-dimensional space (in fact, $2^{\aleph_0}$ dimensions), you get one of the standard constructions of the Stone-Cech compactification of $\mathbb R$. Roughly speaking, it separates, into different points at infinity, all of the possible "ways to go to $\infty$" in $\mathbb R$.
The analogous story works for discrete spaces $X$ in place of $\mathbb R$ (except that I don't need to say "continuous" because all functions on a discrete space are continuous). A point in the Stone-Cech remainder of a discrete space $X$ should be thought of as a "way to go to $\infty$" in $X$. But how can such "ways" be described?
Well, in any compactification, each point $p$ at infinity is in the closure of the original space $X$, and so we can describe its location relative to $X$ by the trace on $X$ of its neighborhood filter, i.e. by $\mathcal F=\{U\cap X: p\in\text{interior}(U)\}$ (where $U$ refers to subsets of the compactification). Note that $\mathcal F$ can't contain any finite subsets of $X$, because such subsets are closed in the compactification (as in any $T_1$ space) and thus disjoint from suitable neighborhoods of any point $p$ at infinity.
For the Stone-Cech compactification of a discrete space $X$, this filter $\mathcal F$ must have one additional property, namely that we cannot have two disjoint subsets $A,B$ of $X$ both meeting all the sets in $\mathcal F$. The reason is that then "going to infinity" in $A$ and in $B$ would be two different ways to go to infinity, both leading to the same point $p$.
This additional property of the filter $\mathcal F$ is equivalent to saying that $\mathcal F$ is an ultrafilter. So this is how ultrafilters enter the picture of Stone-Cech compactifications of discrete spaces.
One then ordinarily continues by saying that, (1) since the filter $\mathcal F$ associated with any $p$ at infinity is an ultrafilter, and different $p$'s must correspond to different ultrafilters (in order for the compactification to be Hausdorff), we might as well identify the points $p$ with the ultrafilters $\mathcal F$, and (2) for the sake of uniformity, we might as well identify the principal ultrafilters (which haven't been used yet) with the points of $X$. With these conventions, one can prove (using compactness) that every ultrafilter on $X$ gets identified with a point in the Stone-Cech compactification of $X$. So the Stone-Cech compactification of a discrete space can be identified with the set of ultrafilters on $X$. Finally, one should verify that the topology is necessarily the one you described.
Best Answer
Since $X$ has the discrete topology, $\lim(F)$ doesn’t exist unless $F$ is a principal (fixed) ultrafilter. However, $\lim(f_*F)$ does exist, since $f_*F$ is an ultrafilter on the compact space $K$, and you do indeed want to define $\bar f(F)=\lim(f_*F)$. Let $U$ be an open nbhd of $\bar f(F)$ in $K$; then $U\in f_*F$, so $f^{-1}[U]\in F$. Let $\mathscr{U}=\{G\in\beta X:f^{-1}[U]\in G\}$; $\mathscr{U}$ is an open nbhd of $F$ in $\beta X$, and it only remains to show that $\bar f[\mathscr{U}]\subseteq U$, i.e., that if $f^{-1}[U]\in G\in\beta X$, then $\bar f(G)\in U$. But $\bar f(G)\in U$ iff $U\in f_*G$ iff $f^{-1}[U]\in G$, so this is clear.