Our story begins with good king Athamas (son of Aeolus, descended from Deucalion).
Athamas married Nephele, goddess 'Cloud', she bore to him Phrixus and Helle and then flew off to the heavens.
Ino (Athamas’ 2nd wife) proved to be a wicked stepmother and contrived to do away with Phrixus and Helle, rigging an oracle to demand them in sacrifice. But Nephele sent a flying golden ram to carry them to safety. Helle fell off (and gave her name to the 'Hellespont')
Phrixus arrived far to the northeast in Colchis (on the shore of the Black Sea) where he sacrificed the ram and dedicated the fleece, in a sacred grove guarded by a great serpent.
In the next generation the grandsons of Aeolus, Aeson and Pelias, were rivals for the throne of Iolcus (in Thessaly, near Athamas' old kingdom). Aeson was the rightful heir but Pelias seized the throne.
Jason is the son of Aeson, brought up in the wild by the good centaur Chiron. Upon reaching manhood, he journeys to Iolcus to claim the throne. He arrives with only one sandal, a sign that gave warning to Pelias.
Pelias therefore demanded that he prove himself by fetching the golden fleece.
Jason assembles the famous crew of the Argo, including: Castor and Polydeuces (sons of Zeus and Leda), Zetes and Calais (flying sons of the North Wind), Telamon and Peleus (fathers of heroes that would later fight at Troy, Ajax and Achilles). Not to mention Heracles (and in one tradition, "not without Theseus")
The main stops on their voyage to Colchis were
(a) the island of Lemnos, where they were welcomed by the Lemnian women (notorious for murdering their husbands).
(b) the land of Phineus, blind prophet beset by Harpies (bird-women who snatched his food). For ridding him of the Harpies, Phineus gave the Argonauts prophecy warning of perils ahead...
(c) the Clashing Rocks, rocking in a narrow passage of the sea, crashing together on anything that passes within.
Having narrowly dodged the Clashing Rocks, they arrive at Colchis, where king Aeetes gives Jason yet another death-defying challenge: he must yoke a fire-breathing bull and plow into the earth the teeth of a dragon (just as Cadmus did). He is then confronted with an army of men sprung from the dragon's teeth; he defeats them (as Cadmus did) by casting a stone in their midst.
But he must still get past the great serpent that guards the fleece. By the most famous version, he does this with the help of Medea, Aeetes' sorceress daughter, who had fallen in love with Jason.
Having taken the fleece they make their escape on the Argo; Medea butchers her own brother to delay their pursuers.
After a fantastic voyage (through Europe and N. Africa) they arrive back in Iolcus. But Pelias refuses to give up the throne. Medea brings about his death by convincing his daughters they can rejuvenate him with magic.
So Jason and Medea are driven into exile and find refuge in Corinth. Jason soon arranges a marriage with the princess, which enrages Medea. She then contrives the death of the princess and the king of Corinth; and when Jason comes to save his sons, she has already murdered them (and refuses even to give him the bodies!)
She then takes refuge at Athens, where she bears a son to Aegeus. She tries to trick Aegeus into killing Theseus, when he arrives incognito; but luckily Aegeus recognizes his sword that Theseus brought.
Medea and son Medus flee at last to Asia where they found a kingdom (of the 'Medes').