Of the Rings of Power and the Third Age is your source for all of this.
Before we begin, it's important to note that there is not actually any difference between the Seven and the Nine. There's absolutely nothing in Tolkien to indicate such a difference, and in fact it seems clear that the effects of them were different based on the species they were given to, rather than anything to do with the Rings themsleves.
In other words, one of the Nine, if given to a Dwarf, would have the same effect on him as one of the Seven. And one of the Seven, if given to a Man, would likewise have the same effect on him as one of the Nine.
So it's actually more correct to speak of the One, the Three and the Sixteen.
Secondly, Tolkien doesn't describe lists of powers like a D&D manual so you're not going to get anything so specific. You also won't get anything such as powers of individual Rings; so far as Tolkien described things, the Sixteen all had the same capabilities, as had the Three (although individual Rings may differ in degree of might, so you'll read about Elrond's Ring being described as "mightiest of the Three", for example).
On to the Rings.
The first set of Rings of Power that the Elves made (I'm excluding the lesser rings Gandalf refers to in Shadow of the Past here) are described as having been made with the following motivation:
"Alas, for the weakness of the great! For a mighty king is Gil-galad, and wise in all lore is Master Elrond, and yet they will not aid me in my labours. Can it be that they do not desire to see other lands become as blissful as their own? But wherefore should Middle-earth remain for ever desolate and dark, whereas the Elves could make it as fair as EressÎa, nay even as Valinor? And since you have not returned thither, as you might, I perceive that you love this Middle-earth, as do I. Is it not then our task to labour together for its enrichment, and for the raising of all the Elven-kindreds that wander here untaught to the height of that power and knowledge which those have who are beyond the Sea?"
It was in Eregion that the counsels of Sauron were most gladly received, for in that land the Noldor desired ever to increase the skill and subtlety of their works. Moreover they were not at peace in their hearts, since they had refused to return into the West, and they desired both to stay in Middle-earth, which indeed they loved, and yet to enjoy the bliss of those that had departed. Therefore they hearkened to Sauron, and they learned of him many things, for his knowledge was great. In those days the smiths of Ost-in-Edhil surpassed all that they had contrived before; and they took thought, and they made Rings of Power.
Thus the objective of the Rings of Power was to assist the Elves in making Middle-earth as fair as Valinor. This is described in various places (e.g Letter 131) as a second fall of the Elves; they wished to have the bliss of Valinor but yet remain in Middle-earth as the greatest of the incarnates. Letter 131 further goes on to make it explicit how the Rings would achieve this objective:
The chief power (of all the rings alike) was the prevention or slowing of decay (i.e. 'change' viewed as a regrettable thing), the preservation of what is desired or loved, or its semblance – this is more or less an Elvish motive. But also they enhanced the natural powers of a possessor – thus approaching 'magic', a motive easily corruptible into evil, a lust for domination.
But of course there is a dark side:
And finally they had other powers, more directly derived from Sauron ('the Necromancer': so he is called as he casts a fleeting shadow and presage on the pages of The Hobbit): such as rendering invisible the material body, and making things of the invisible world visible.
The Three Rings of the Elves are called out in Rings of Power as having additional capabilities:
Now these were the Three that had last been made, and they possessed the greatest powers. Narya, Nenya, and Vilya, they were named, the Rings of Fire, and of Water, and of Air, set with ruby and adamant and sapphire; and of all the Elven-rings Sauron most desired to possess them, for those who had them in their keeping could ward off the decays of time and postpone the weariness of the world.
If you're observant you'll have noticed something odd: Tolkien, in Letter 131, notes that prevention of decay was a power that all the Rings had, but yet in Rings of Power he states that the Three were particularly desirable on account of this ability. But how could the Three be so desirable if all the Rings had this power anyway? There's absolutely nothing I'm aware of in his writings that reconciles these two statements, so we'll just have to live with it.
And that's the extent of what we know about the powers of the Rings based on Tolkien's writings. I haven't given every single quote (of course), but the other mentions of them don't contain any further information.