I will summarize the reading and add my comments and speculations in [] brackets. Do not repeat or take the stuff in brackets as fact.
Aegon I did indeed have only two children, Aenys (born in 7 AC) by Rhaenys, and Maegor (born in 12 AC) by Visenya.
[I might be partially responsible for the confusion of Aegon's alleged daughters, since I found - and still find - it somewhat strange that Aegon should only have two children by his two sister-wives despite the fact that he was already married to them prior to the Conquest. Back when GRRM read 'The Sons of the Dragon' for the first time, I asked the attendants of the Con very specific - and perhaps suggestive - questions. This may have actually confused and obscured things...]
Aenys Targaryen was indeed a sickly and weak child, with spindly arms, watery eyes. He always wept, and refused to be fed by midwives which is why Rhaenys was forced to feed him almost exclusively herself [that could be an explanation why Aenys did not get a little sibling after his birth - women do not conceive while they are still breastfeeding].
Rhaenys dies when Aenys is three years old (i.e. in 10 AC). This event apparently shatters young Aenys - he cries all the time, and his father grows desperate. There are doubts whether the boy will live. Aegon is asked to take another wife, because Visenya is supposed to be barren. Apparently many young women are presented to the king, but Gyldayn doesn't really know what's happening there, since Aegon eventually refuses to remarry. In 11 AC, Visenya is pregnant, and announces that the child will be a very strong boy long before he is actually born. At his birth, Maegor weighs twice as much as Aenys - he is a big and healthy boy.
[Speculation: We know that Visenya was supposed to be into magic and the black arts - my guess is that she was indeed barren - or had very serious troubles to conceive (especially since she was about 40/41 in 11 AC) - and used a spell to ensure that she did conceive a male child. She would have done this, at this point, most likely to give Aegon another son to ensure the survival of the dynasty - not because she was overall ambitious. Visenya does not strike me as a woman who was all that keen to be a mother.]
Aenys seems to bond with his dragon, Quicksilver, shortly after the death of his mother (i.e. while he was 3/4). After that, his condition improved, 'and as the dragon grew, so did Aenys'. Due to the fact that Aenys was such a weak and sickly child, rumors circulated that this boy could not have been fathered by Aegon the Conqueror - who was a very manly warrior - but rather by one of the many singers, dancers, and mummers who surrounded Queen Rhaenys before her untimely death. Those rumors died when Aenys bonded with Quicksilver. [History repeats itself - when Rhaenyra's three elder son all bond with their hatchlings, this is taken as a sign that they are legitimate. Nobody seems to utter aloud that this may just work because they have Targaryen blood through their mother - as would Aenys through Rhaenys. The thought that all the Targaryens do not, in fact, descend from Aegon the Conqueror would be quite funny...]
The brothers were not close as children. [I imagine the age gap would have ensured as much, even if the other factors hadn't been there.] Aenys was raised by Aegon I, and trained by the knights of the Kingsguard. As Heir Apparent to the Iron Throne, he accompanied his father always on his royal progresses, resided with him in KL when the king was there. [It seems that the early Targaryen kings were essentially traveling kings, keeping the vast Realm together the same the early kings/emperors of th Holy Roman Empire did - on horse back! This would have changed only, I think, during the reign of Jaeharys I.]
Aenys loved songs, was a fine singer himself, was clever - although not in a scholarly way - and liked to ride. His favorite mount was, of course, his dragon, Quicksilver. Aenys was no great warrior/knight, but did also not disgrace himself in tourneys. [I guess he would have been somewhat better than Prince Valarr in THK, but not all that much.] He was also interested in alchemy, and other obscure things. Aenys could also make friends very easily, and was very popular with the women.
Aegon occasionally gave Blackfyre to Aenys to train with the sword.
Maegor, on the other hand, resided with Visenya mostly on Dragonstone, and thus became in time known as 'the Prince on Dragonstone'. [It's obvious that the title 'Prince of Dragonstone' derives from this fact, and was originally not the title of the Heir Apparent. We'll later learn how and when that changed.] He was trained by the master-of-arms of Dragonstone, a knight of House Corbray [Didn't catch the first name, perhaps somebody else got it.]. Maegor had no true friends, and sadistic tendencies (he supposedly killed a cat in a Joffrey-like fashion, and finished off a horse who kicked him). In 25 AC - when he was 13 - Visenya gave Dark Sister to him [For reasons I can't discuss here, this is strange, but I guess Maegor returned the sword to his mother after Aenys gave him Blackfyre.] Maegor's skills with weapons were unmatched.
Now I have to stop, since I'm actually at this Con to have fun, and not to write down stuff which I can just as well write down when I'm back at home.