The forging of the Rings of Power was always the most likely place for this story to start, as it's the strongest connective tissue between the Second Age and the events of Lord of the Rings (which takes place 4,859 years after the forging of the One Ring). In this story, the Dark Lord Sauron, disguised as the elven prince Annatar, goes among the elven-smiths of Eregion, who live in the shadow of Moria, and corrupts them with promises of riches and power. He tricks them into forging the Seven and Nine Rings of Power to corrupt dwarves and men, but is taken by surprise when the elven master-smith Celebrimbor forges the Three Rings for the elves. Sauron then uses the knowledge he has gained to forge the One Ring itself in the fires of Mount Doom, betraying the elves and leading to a devastating war with Sauron's forces. Established characters like Elrond, Galadriel, Celeborn and possibly Thranduil (the father of Legolas) play roles in this conflict.
The Amazon presentation doesn't feature any footage from the series - because it hasn't shot a single frame yet - but it does feature a shot of the iconic One Ring, glowing with its famous inscription.
With a five-season run planned for the series, it's likely we will see further seasons exploring later periods of the Second Age, including the Downfall of Númenor and the War of the Last Alliance.