The first objectively clear instance of Rand hearing a voice and consciously associating said voice as belonging to someone besides himself occurs in The Fires of Heaven, at the end of Chapter 2:
Ilyena never flashed her temper at me when she was angry with herself. When she gave me the rough side of her tongue, it was because she . . . His mind froze for an instant. He had never met a woman named Ilyena in his life. But he could summon up a face for the name, dimly; a pretty face, skin like cream, golden hair exactly the shade of Elayne’s. This had to be the madness. Remembering an imaginary woman. Perhaps one day he would find himself having conversations with people who were not there.
(The bold emphasis is my own.)
Immediately prior to the passage cited above, Rand unthinkingly addresses Moraine as "little sister", and immediately wonders where the words came from. During the same conversation, Rand declares that he will have peace or be buried in the Can Breat, whatever that is.
There are of course earlier episodes a this that are often attributed to LTT, with supporting evidence ranging anywhere from purely-speculative to all-but-certain, but by and large the potential instances from TSR are much more subjective.
While the disembodied voice of Lews Therin inside Rand's head inspires the most speculation and debate amongst readers, I think it's just as interesting to take a critical look at all of the other symptoms of mental illness Rand develops over time.
Rand spends the majority of TEotW terrified, overwhelmed, exhausted, and isolated, with occasional brief delusional episodes induced from channeling thrown in for spice. By the time of TGH, Rand has matured a lot emotionally, and he is probably at his peak mental health for the entire series during this time.
We see a lot less of Rand in TDR, but what little we do see is quite telling. He exhibits clear signs of depression in the beginning when he disproportionately blames himself for failing to stop the trollocs, full blown mania in the middle when unhesitatingly decapitates a group of darkfriends and goes so far as to manipulate the corpses by having them bow down, and finally delusions of grandeur at the end when he convinces himself that he killed a deity and declares himself a messiah and savior of humanity. By the end of TDR, Rand is without a doubt suffering from persistent mood disorders, but all signs of psychosis up to this point seem to have been transient and nonrecurring.
This brings us to The Shadow Rising. The first two POVs we have with Rand are the bubble of evil scene and his first romantic scene with Elayne. In Rand's third POV, he is approached by Lanfear. I'm now convinced that it is during this confrontation that we see the very first tangible manifestation of Lews Therin's memories leaking into Rand's mind:
“And you loved power!” For a moment he felt dazed. The words sounded true—he knew they were true—but where had they come from?--TSR,Ch.9
Minutes later the trolloc attack begins. Exacerbated by the wildness-inducing effects of Callandor, I think Rand's desperation to stop the trollocs induces a full-scale dissociative break from reality, starting here:
Now. The thought floated like cackling laughter on the rim of his awareness. He severed the flows rushing out of him, leaving the thing still whirling, whining like a drill on bone. Now.
Those 'now's may very well be the birth of the LTT persona Rand eventually attaches to the memories he has received.