Basic subject/predicate: their stories tell
“Taken together” is an adjectival phrase that modifies “their stories”, indicating that the stories as a lump sum give insight that one could not get by reading only one or a few of the stories.
tell not only of the tragedies and rare triumphs of the day....:
This is a bit tricky; I would suggest that this is an object complement, with an implied direct object, “a story”. With the object included, it would read like this:
tell not only [a story] of the tragedies and rare triumphs...
The rest of this (“and rare triumphs”, etc) are just standard conjunctive and adverbial phrases completing the thought, like the following paraphrase:
(This compendium of individual accounts) does not merely tell about the individual struggles, losses and victories during the immediate time period....
but also of the tumultuous decade that followed.
Correlative conjunction continuation following “not only”: what may seem more tricky, though, is that this is also the second part of the object complement “tell [a story]...of the tumultuous decade ahead”.