Example of countably infinite sets that is connected?

First of all, it is meaningless to talk about a set being "connected" or "totally disconnected". These words only have meaning when you are talking about a topological space, not just a set.

As for a countably infinite connected topological space, there are plenty of such spaces. The easiest example is to take any countably infinite set $X$ and give it the indiscrete topology, where the only open sets are $X$ and $\emptyset$. Intuitively, you can think of this as a "space" where all the points are right on top of each other, so you can't tell any of them apart (but there are nevertheless infinitely many of them).

More surprisingly, there exist countably infinite connected Hausdorff spaces. See https://mathoverflow.net/questions/46986/countable-connected-hausdorff-space for some examples.

A simple example, which is a $T_1$ space, but not $T_2:$ Let $X=\mathbb N$ and let $T$ be the co-finite topology on $X$ : Any non-empty $S\subset X$ is open iff $X$ \ $S$ is finite.

It is connected, as any two non -empty open sets have infinite intersection. And if $Y$ is any infinite subspace of $X$ then $Y$ is homeomorphic to $X$, so $Y$ is connected. And if $p\in V\subset X,$ where $V$ is a nbhd of $p$, then $V$ is connected.