The original incarnation of this team was a 31st century counterpart to the Avengers, similar to DC's 30th century superhero team, the Legion of Super-Heroes. They are most well-known for their role in the Avengers' Korvac Saga. Although there were many members of their team, the only ones relevant to the current team or upcoming film are Major Victory (a 31st century counterpart to Captain America) and Yondu (a blue-skinned archer-warrior from Alpha Centuri).
Created in 1969, the Guardians finally had their own series in the mid-90s by Jim Valentino, but it did not catch on and the characters were soon forgotten.
The inaugural event that preceded the debut of the team consisted of a prologue issue with four miniseries including Nova, Silver Surfer, Ronan, and Super-Skrull followed by the Heralds of Galactus mini-series. Beginning in Thanos and the Drax the Destroyer miniseries, this event features the invasion of bug-like entities from a sector of space called "The Crunch" where the Negative Zone meets the "Positive" universe. Led by Annihilus, the unstoppable swarm is dubbed "the Annihilation Wave."
In their first strike, the Annihilation Wave destroys an intergalactic prison known as the Kyln where a half-human superhero who called himself "Star-Lord" is being imprisoned following his destruction of a small planet to save many others. In their second strike, the Annihilation Wave destroys Xandar and the galactic peacekeepers known as the Nova Corps leaving only Centurian Richard Rider with their entire power and legacy.
The event comes to a conclusion in the Annihilation miniseries as the intergalactic powers form a united front, kill Annihilus, and force the invading army to retreat (though several planets remain occupied).
Following the aftermath of Annihilation, Nova received his own series focusing on Richard Rider baring the burden of the entire power of the Nova Corps and the Xandarian Worldmind. This series ran along side Guardians of the Galaxy and was also written by Abnett and Lanning, so while it is not necessary to read one to enjoy the other, they do go well together.
This series focused on the struggles of Richard Rider following the death of the Nova Corps. With his former team, the New Warriors, now hated on Earth following Civil War, he finds he has no place on Earth, but instead struggles to be a one man Corps.
Annihilation: Conquest (2007-2008)
This event crosses with the Nova ongoing series, but is also told in the miniseries Annihilation: Conquest - Quasar, Wraith, and Star-Lord. As Star-Lord helps rebuild the Kree Empire following Annihilation, he unknowingly helps to facilitate a Trojan horse invasion led by the techno-organic Phalanx. Eager to atone for his most recent mistake, Star-Lord leads a team of elite Kree convicts into a dangerous mission in the heart of Kree space. This is the team that would soon become the basis for the Guardians of the Galaxy including Bug, Rocket Raccoon, Mantis, Groot, and Captain Universe.
Guardians of the Galaxy (2008-2010)
Following the battle in Annihilation: Conquest, Star-Lord reforms his team as a proactive group designed to prevent the next universe damaging events. As a result of the previous storyline, the fabric of space/time has been weakened and their first priority is to repair the places where reality has been stretched to the breaking point. The critical component in their plans is the all-powerful Adam Warlock, who is a human who has evolved to the point where his connection to quantum energies is indistinguishable from magic.
Although Bug and Captain Universe were not in the initial lineup, they were replaced by Adam Warlock, Quasar (Phylla Vell), Gamora, and Drax the Destroyer. After discovering the severed head of a Celestial floating in a space just outside of the universe, the Guardians are able to use the Celestial head as a teleporter with unlimited range. This place, referred to as Knowhere, is used as an intergalactic port that barely tolerates the Guardians' presence. Their security chief is a telepathic dog with a Russian accent known simply as Cosmo.
Secret Invasion (2008-2009)
Although not directly tied to these events, this Marvel event featured the invasion of Earth by alien shapeshifters. This required both the Guardians and Nova to be preoccupied on vaguely related adventures. Yet this event directly leads into the following cosmic event...
When the Skrull invaded Earth, they replaced (amongst many) the Inhuman king, Black Bolt. Unwilling to led this threat pass, Black Bolt converted the Inhuman capital city into the starship that it originally was and led his people to destroy the fleeing invaders. After they were destroyed, the ship turned to Hala, the home planet of the Kree Empire. The Inhumans and the Kree have been enemies since the Kree created the Inhumans with genetic experimentation designed to help fix their own genetic stagnation. Surprisingly, the Kree ruler, then Ronan the Accuser, surrendered to the Inhuman invaders with the belief that their strength would be to the benefit of the war-weary empire.
Meanwhile, the other great empire of the known universe, the Shi'ar Empire, had been taken over by a human mutant - Gabriel Summers who preferred the name Vulcan. An omega-level mutant, Vulcan possessed the ability to manipulate limitless energy as well as a sadistic desire for power. Vulcan had taken the Shi'ar throne through marriage, deceit, and power, killing both his own father and the Shi'ar Emperor D'Ken in the process.
Soon, Vulcan's expanding Shi'ar Empire meets the recently conquered Kree Empire and the two empires engage in a war that threatens the fabric of reality, forcing the Guardians and Nova to find a way to bring the war to a swift and stable conclusion.
Realm of Kings (2010)
Following the conclusion of War of Kings, both Black Bolt and Emperor Vulcan are missing and presumed dead as the Inhuman doomsday weapon caused a massive interstellar phenomenon called "The Fault." Although nearly suicidal to enter, the various universal representative each send in their explorers only to discover a universe where death has been destroyed and life has been perverted. Seeking to invade their world, they call it the "Cancer-verse."
The Thanos Imperative (2010)
Following the conclusion of both Nova and the Guardians of the Galaxy, these two titles have their respective stories resolved in the dramatic conclusion to this cosmic epic of events from creators Abnett and Lanning. This series features the battle of our universe against the Cancer-verse as the Guardians use Thanos, avatar of death, to combat a universe where death doesn't exist.
This event ties up all of the cosmic events since Annihilation.
Annihilators [Earthfall] (2011)
Following The Thanos Imperative, the telepathic dog from Knowhere, Cosmo, recruits a team to replace the Guardians in order to fulfill Star-Lord's dying request. No longer a team of unpredictable rebels, Cosmo is directed to recruit the most powerful - the Annihilators. This team consists of Silver Surfer, Ronan, Gladiator, Quasar, and the unanticipated addition of Spaceknight Ikon.
The Annihilators are featured in two miniseries. The first self-titled series features the Annihilators versus the Dire Wraith as they ally themselves with the Spaceknights. The second series (subtitled "Earthfall") features an encounter with the Avengers as the Universal Church of Truth resurrect Adam Magus on Earth.
Guardians of the Galaxy - Marvel NOW! (2013-present)
The most recent incarnation of the Guardians of the Galaxy is written by Brian Michael Bendis. There is a significant change in the character of Star-Lord. In addition to being blonde where he was previously brunette, there is a focus on Peter's promiscuity and issues with his abandoned father, who apparently is actually ruler of Spartax. This series also includes Iron Man and recent Marvel acquisition Angela to the team.
In my estimation, the Bendis run is not nearly as good as the Abnett and Lanning run, and I would not recommend it. The title takes a major shift in tone and story as well as stunt-casting members like Iron Man, Angela, and Venom who dominate the storyline.
If you want to just dive into it all (as I did), I recommend that you start with Annihilation and end with Thanos Imperative. Don't avoid the events as they are crucial to the story. You can skip Secret Invasion, but if you like Guardians, you'll probably also like Nova and they work well as companion titles.