Given the regular set $S = a^*ba^*ba^*$, is the set $S'$ of all first thirds of strings in S (with length divisible by 3) regular?
The first step is figuring out the language of first thirds of $S$. Let us denote this language by $L$. A good starting point is to consider the language $P$ of prefixes of $S$, since clearly $L \subseteq P$. The language $P$ is easy to find: it is
$$
a^*ba^*ba^* + a^*ba^* + a^*.
$$ 
With some work, you can show that $L$ and $P$ are rather close to each other, and so $L$ is regular. Details left to you.



More generally, if $S$ is any regular language, then the language $L$ of all first thirds of words in $S$ is always regular. The easiest proof of this is via NFAs. You can take it as a challenge, or look up material on the very similar operation of taking the first half of all words in a regular language.