None of the Rings of Power were originally intended for Men. They were all meant for Elves. Sauron’s plan was to ensnare them and rule them through the Rings of Power.
By the time he left Eregion, he had helped them forge 16 rings: the Seven and the Nine. That implies these rings were made in two sets: the Nine first, then the more potent Seven. After he left, Celebrimbor and the other Mírdain forged the Three alone. Elrond describes the Three negatively: “not … weapons of war or conquest”, which would seem to describe the Nine; “not [for] strength or domination or hoarded wealth”, which might be the powers of the Seven. Then he describes the Three positively: for “understanding, making, and healing, to preserve all things unstained.” The things he describes negatively are attributes Sauron valued: he’d find them useful in his servants. It’s difficult to imagine how “understanding, making, and healing, to preserve all things unstained” would be of any interest to Sauron, “Base Master of Treachery …, faithless and accursed”.
The greatest of the Seven was given to the King of Khazad-dûm: the Dwarves claimed it was given by Celebrimbor, his friend and ally; later, the Elves were not certain. I think the gift of Celebrimbor more likely, since he was probably trying to hide them from the advancing army of Mordor. Nenya he sent to Galadriel, whom he and the Mírdain had driven out of Eregion after Sauron-Annatar cozened them; Narya and Vilya he sent to Gil-galad. It made sense for him to keep the other 15: while the Noldor didn’t dare wear them while Sauron wore the One, they might be still find a way to make them useful as weapons of war.
Sauron’s doling the remaining six of the Seven to the other Dwarf-houses could be wicked inspiration from Celebrimbor’s giving one to Durin’s Folk. Three of the Nine went to Númenóreans, leaving six to ensnare king and warriors from other nations of Men, though I suspect the Númenóreans were ensnared last: it pays to know how the Rings affect other races, and Sauron could raise up his own allies (Khamûl the Black Easterling is the only Nazgûl Tolkien names) before trying it on your enemies. Men probably fell quickly, as Gandalf told Frodo: “sooner or later – later, if he is strong or well-meaning to begin with, but neither strength nor good purpose will last – sooner or later the dark power will devour him.” The Dwarves proved more frustrating; it is interesting that the Ring of Durin’s Folk was the last recovered, just as it was the first given.
But we are no closer to why there are Seven and Nine rings in two separate sets. There are Seven Vices as opposed to Seven Virtues: maybe the number Seven is somehow related to that notion. But Nine? The US Army uses “nine principals of war” (Objective, Offensive, Mass, Economy of Force, Maneuver, Unity of Command, Security, Surprise, and Simplicity), but it is unlikely Tolkien, an Englishman, was referring to that. For what other reasons he might have chosen Seven and Nine for his numbers of Rings?