Not surprisingly, the 1963 Jason and the Argonauts movie looks very old-fashioned, particularly in photography and in what constitutes a buff male character, although Ray Harryhausen's stop motion animation process special effects still look pretty good 46 years later. The film is worth seeing if the story of Jason and the Argonauts interests you, but not if you want the story of Medea.
Medea is brought in earlier in the movie than in the Argonautica of Apollonius of Rhodes, which covers approximately the same sequence of events. Chaffey's version's first sighting of Medea is when the Argonauts are still en route to Colchis. She is a beautiful, shipwrecked, damsel in distress. Her character develops no further.
In the Argonautica, as in the movie, Medea is a priestess of Hecate. In the Argonautica, she is also the daughter of the king and the sister of Chalciope. In the movie, Medea sits back to watch Jason win the fleece on his own, in accord with a devaluation of women one would have thought more typical of the third century B.C., which is when Apollonius wrote his Argonautica.
Apollonius' Medea falls for Jason, thanks to the love god, but her decision to help Jason at the expense of her father is based partly on other familial concerns, specifically, the sons of her sister, Chalciope.
Jason, as a suppliant, recalling how useful Ariadne had been to another hero, Theseus implores Medea to help him, in the Argonautica.
Medea then gives him a charm and tells him what he must do to fight the sown men who will come after him. [Argonautica Book III. 1026-1062.] Jason says he will take her away and marry her.
The Argonauts win the sown men contest and then the next day Medea asks that Jason honor his promise to marry her and take her away. After Jason publicly promises to do so again, Medea takes him to get the fleece. Medea charms the monster guarding it, while Jason hangs back in fear. When Medea instructs him to, Jason grabs the fleece and they head towards the ship, where for a third time, Jason says he will take Medea home to become his wife. All this is missing in the movie, since its focus is firmly fixed on showing how great a hero Jason could have been.