I am trying to figure out the missing rests in the time signature. I think for the first bar there will be a dotted quarter rest. I am not too sure about the next one. Any hints or directions appreciated.
In 9/8 time, it is customary to notate it as three groups of three beats each. Thus, you are correct that the first measure would be completed by a dotted quarter rest.
Keeping this convention in mind, and observing that the second measure's "middle" three beats are accounted for by the dotted quarter-note, should allow for the question to be answered.
9/8 is equivalent of 9 eighth notes per bar. BUT because the top number is a multiple of 3, those eighth notes are set out in groups of 3 (or equivalents of 3 quavers). As in, the bar could be divided into 3 equal parts.
It's the same with other compound time signatures - 6/8 divides into 2x3, 12/8 into 4x3.
So, now, it's pretty simple sums!