Channeling sickness

Channeling sickness is a condition experienced by people born with the ability to channel, but who are unguided in learning how to do so. As such, these wilders unintentionally and unknowingly use the One Power unsafely, which generally causes a slow, sapping illness that eventually results in death. If a trained channeler reaches a person with the channeling sickness in time, the afflicted person can often be saved.

Moiraine's explanation (to Nynaeve)

“The age varies, but always comes young – there was something you wanted more than anything else in the world, something you needed. And you got it. A branch suddenly falling where you could pull yourself out of a pond instead of drowning. A friend, or a pet, getting well when everyone thought they would die. 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.”

...

“Each time, the reaction comes closer to the actual touching of the Source, until the two happen almost together. After that there are no more reactions that can be seen, but it is as if a clock has begun ticking. A year. Two years. I know one woman who lasted five years. Of four who have the inborn ability that you and Egwene have, three die if we do not find them and train them. It is not as horrible a death as the men die, but neither is it pretty, if any death can be called so. Convulsions. Screaming. It takes days, and once it begins there is nothing that can be done to stop it, not by all the Aes Sedai in Tar Valon together.”

Rand's development

First time

On the way to Taren Ferry, Rand touches the True Source for first time, cleansing Bela from tiredness.

Just a week later in Baerlon at the Stag and Lion inn, Rand wakes up with a terrible headache; later he feels like he has a cold, and feels just odd, drifted. When he meets with Whitecloaks he behaves weirdly, aggressively, feverishly, dangerously.

Second time

After being chased by Trollocs from Shadar Logoth, Rand causes a boom on Bayle Domon's ship to knock a Trolloc who was about to kill Rand overboard. Four days later, he climbs atop the mast of the ship: legs splayed to either side of the mast, nothing holding him to his precarious perch but his two hands on the stay, he laughed. Gulping huge breaths of the fresh, cold wind, he laughed with the exhilaration of it. Thom then says "Burn me, boy, you've got everybody thinking you've gone mad."

Third time

Four Kings, The Dancing Cartman – Rand kills Darkfriends and blasts open a wall using lightning.

Mat and Rand reach a small village with the inn The Queen's Man and right after they agreed to perform for their stay, Rand gets sick in the kitchen. Shivering strongly, teeth chattering, bones feeling as if they were freezing. Shaking head to toe. After freezing he feels being hot as if he were in an oven. Chill then fever, chill then fever. A few hours later he feels much better, wondering how did he get over being sick so fast. As if it must have been the gift of the Light.

Fourth time

Tarwin's Gap – Rand kills Trollocs, Draghkar, Myrddraals with Fire and Earth then uses the Power against Ba'alzamon for the first time.

This time the fever comes right after he channeled. "When he awakes, he can't think or remember. [...] Pain comes next and the memory of raging fever and bruises where shaking chills had thrown him around like a rag doll.".

After these there is no mention about him being sick after channeling.

Rand dies within two years of his first channeling attempt, but not because of the sickness. Most probably Asmodean's teaching can be considered as the saving guidance. Rand is of course also ta'veren, which helps his luck.