In short, they were originally intended for elven leaders, but eventually they were given to the rulers and kings of men and dwarves. Hobbits had no rulers, and they were barely noticed most of the times.
There are a total of 20 rings, 4 of which forged in secret. The first 16, were some sort of practice, and the Elven Smiths, Celebrimbor, and Sauron worked on them. Sauron used these opportunity to bind them to his own ring of power secretely, which he forged in mount Doom. These 16 rings were in elven hands, and meant to be worn by elves. Then, Celebrimbor alone created 3 elven rings of power, not binded to the one ring.
When Sauron put on his ring of power, the Elves sensed it, and removed theirs. Sauron was mad, for his plan had failed, and entered war on the elves, demanding the return of the rings. The elves managed to hide the 3 most powerful rings, but Sauron eventually retrieved the other 16.
He decided to give them now to men, and dwarves, who he believed would bow to his will more easily. The 9 kings, at first drew great power from the ring, but eventually became Nazgul. The dwarves, at first archieved great riches, but the rings just made them grow more greedy, and that lead them to their ruin.
The 3 elven rings were passed on. Gandalf, Galadriel, and Elrond were the bearers of them by the time of LOTR.
What was Sauron's original plan? It's hard to tell. We can't say if he intended to create more rings of power after using the original 16 for controlling the elves. We can't tell if he intended to have more made for humans and dwarfs, or if he simply thought elves would suffice. His plan fell through though, and he had to improvise. Hobbits didn't have a great weight on Middle Earth though. There were no leaders to control or target.