Countability union of all finite and countably infinite sequences over finite alphabet Is the set of all finite and countably infinite sequences over $\{0,1\}$ countable?

From my analysis, I think it is countable. I think of this as the set of all strings from a finite alphabet $\Sigma=\{0,1\}$, hence $\Sigma^*$. (is this a good assumption?).

I later show that I can count each string in the following order: $0$,$1$ (length $1$), $00$, $01$, $10$, $11$ (length $2$) and so on.

However, I am very confused from the initial requirement "finite and countably infinite sequences". Does my method account for the countably infinite strings?
Your solution for proving that $\Sigma^*$ is countable is correct.
However the set $\Sigma^\omega$ of infinite sequences is not countable,
since it is in bijection with the set of subsets of $\mathbb{N}$, which is not countable. See this question.