There are two separate components to the series - the stories and the novels. The stories are mostly independent and share just a common structure and fit into the same history, but two sequences have recurring characters: the troubleshooting team of Donovan and Powell, and the great roboticist Susan Calvin. Then the novels feature the human Lije Bailey and the robot R. Daneel Olivaw and go on from there. There are separate cases per novel but the novels are much more closely related to each other than the stories are. The initial publication was
I, Robot (which features a framing story linking all the stories and isn't available in the other compilations of the stories which are also arranged differently). Then he released the first two novels. Then, for some reason, he released
The Rest of the Robots which is a compilation of those two novels along with eight stories not previously collected. These eight stories were later published in a separate volume by Pyramid. After that, he wrote the occasional robot story and collected them like any other stories in his general collections until, finally, in 1982, he released
The Complete Robot which, as I say, dropped the framing story of
I, Robot and rearranged them, the eight stories from
The Rest of the Robots, all the robot stories he'd written between then and 1982, and even a couple of stories that stretch the definition of a "robot story" and weren't originally really treated as such. Then he wrote the last two robot novels, the last of which bridges the robot stories/novels and the Empire/Foundation stories/novels. And, along the way, there were even various other robot compilations such as
The Robot Collection which collects
The Complete Robot and the first three robot novels. (It's kind of ironic that something which
contains something called
The Complete Robot is incomplete in terms of both novels and stories.) After
The Complete Robot and the last two robot novels, he also wrote a few more robot stories - one is only available in what is otherwise a collection of unrelated previously collected non-robot stories called
Robot Dreams, at least three are only available in one which is a collection of related, previously collected robot stories (and non-fiction) called
Robot Visions, and the last few are in an original non-robot collection (half made up of non-fiction) called
Gold.
I'd recommend starting with
I, Robot and then getting
The Complete Robot as well. Then move on to the novels (though you can go to them directly after
I, Robot) either individually or in
The Rest of the Robots. Then, if you're a raving fan by that point, as you surely must be
, then go on and get all of his collections and pick up the rest of the rest of the robot stories that way. However, you can certainly start with
The Complete Robot as well and only miss the framing story and the arrangement.
Hope that helps (though I'm afraid it might be more confusing than helpful).