TL;DR: Rand first channeled while fleeing Emond's Field, in order to help Bela run faster. However, it is his second use of the Power that causes the odd behavior on the boat.
According to Moraine at the end of EotW, she suspects that Rand first channeled when they were fleeing the Draghkar:
The Aes Sedai nodded. "At Watch Hill, Bela had no need of me to
cleanse her of tiredness; someone had already done it. She could have
outrun Mandarb, that night. I should have thought of who Bela carried.
With Trollocs on our heels, a Draghkar overhead, and a Halfman the
Light alone knew where, how you must have feared that Egwene would be
left behind. You needed something more than you had ever needed
anything before in your life, and you reached out to the one thing
that could give it to you. Saidin."
This event occurred in Chapter 11 ('The Road to Taren Ferry'). As the chapter begins, Rand is worried that Bela - Egwene's mount - will not be able to keep up with the rest of the group. Apparently he unconsciously did the trick of wiping away the horse's fatigue so that Egwene would not be caught by the Draghkar or Trollocs following them:
If Bela fell back, he would fall back, too, whatever Moiraine and Lan
had to say about it. Back where the Fade and the Trollocs were. Back
where the Draghkar was. With all his heart and desperation he silently
shouted at Bela to run like the wind, silently tried to will strength
into her. Run! His skin prickled, and his bones felt as if they were
freezing, ready to split open. The Light help her, run! And Bela
Effects of Channelling
As Moraine explains to Nynaeve in Chapter 21, a wilder's untrained use of the One Power results in several various symptoms:
"You felt nothing special at the time, but a week or ten days later
you had your first reaction to touching the True Source. Perhaps fever
and chills that came on suddenly and put you to bed, then disappeared
after only a few hours. None of the reactions, and they vary, lasts
more than a few hours. Headaches and numbness and exhilaration all
mixed together, and you taking foolish chances or acting giddy. A
spell of dizziness, when you tripped and stumbled whenever you tried
to move, when you could not say a sentence without your tongue
mangling half the words. There are others. Do you remember?"
Once the group reached Baerlon, Rand appears to experience these very symptoms - first the headaches & dizziness, later the giddiness & foolish behavior - ultimately resulting in his fearless face-off with the Children of the Light in Chapter 15:
The tingling filled Rand; the heat had grown to a fever. He wanted to
laugh, it felt so good. A small voice in his head shouted that
something was wrong, but all he could think of was how full of energy
he felt, nearly bursting with it. Smiling, he rocked on his heels and
waited for what was going to happen. Vaguely, distantly, he wondered
what it would be.
This behavior in Baerlon ultimately only lasts for a few hours, passing without any further explanation yet leaving Rand - and Mat - wondering at his own reckless behavior.
Later, while boarding Bayle's boat, he again channels to move the boom and knock the pursuing Trolloc overboard. It is this second use of the Power that results in the odd behavior you describe while travelling downriver.