Everyone going into this book should know it's Part 1 of a 2-part tale, and this part ends abruptly.
If you know a lot of classic or canonical literature, you'd recognize the subtle and not so subtle nods to famous dead authors and artists scattered, not without purpose, throughout this book. And if you're not familiar with the classics, not a problem. You can always Google them afterward. Never having read John Keats or John Muir or the Talmud or know the ethics of ecology by heart wouldn't get in the way of your enjoyment of this book.
The Hyperion pilgrimage/adventure starts out with seven strangers chosen specifically for a mission that's a matter of life and death. These seven individuals are to go to the planet Hyperion and journey deep into its depths in search of the Valley of the Time Tombs. Once there, the group is supposed to petition the Shrike, a terrifying legendary creature responsible for countless deaths, but it isn't as mindless or unreasonable as legends say. It's believed the creature, after listening to each person's story, would grant a wish to just one person and then it would kill the rest. See? Not completely mindless or unreasonable.
Before reaching Hyperion, the group decides to share their tales during the journey to better understand one another or perhaps to know what to expect from one another once they meet the Shrike. This is the point in which the story becomes stories within a story. Each one builds upon the previous, and each containing a unique piece of the Shrike mystery.
One more thing. Among the seven strangers, one is a double agent who might or might not be working for the Shrike. Can you spot him or her before the group reaches the Valley? Also, the tales these character weave are not necessarily based on any truth. They could just be making things up on the fly and none would be the wiser, but readers might figure it out before the big reveal. This is where your knowledge of literary trivia comes into play.
The storytellers are:
Father Lenar Hoyt, a sickly Catholic priest whose story is another story within a story about another priest's experience on Hyperion, basically an homage to Joseph Conrad's Heart of Darkness 
Colonel Fedmahn Kassad, a former military personnel better known as a merciless slaughterer; from a young age he's been given a unique "gift," a connection to the Shrike
Martin Silenus, an angry bitter poet who, in the tradition of poets before him, lived a turbulent life of excess and few regrets; he too has a special connection to the Shrike
Sol Weintraub, a historian and philosopher desperately seeking answers and/or a cure for a time-related affliction--not unlike Benjamin Button
Brawne Lamia, a hardboiled private detective and somewhat unusual homage to the noir days of Hammett and Chandler
the Consul, a politician burdened by the weight of his lineage and past, who seeks whatever answers that might be found in the Time Tombs
Het Masteen, a Templar and captain of the treeship Yggdrasill; still a mystery in this first half of the Hyperion adventure
And the adventure continues to be fraught with mystery and doubt.
 Because it's hilarious.
 Which I will never not read as "Time Bombs"
 "Frame story" is what this literary technique is called. Famous examples include: One Thousand and One Nights, Chaucer's Canterbury Tales, Frankenstein, Wuthering Heights, Interview with the Vampire, The Princess Bride, Cloud Atlas, and many more
 Another famous frame story
A few thoughts and reflections: