Their eldest son was Charles, who was made Prince of Viana by his maternal grandfather. He was also named the successor to Navarre after his grandfather and mother, a decision by them which angered John.
When Blanche died in 1441, John refused to allow Charles to rule, retaining control of Navarre, even though Charles was referred to as king by the people of Navarre.
John decided that, instead of naming Charles as the successor to his own possessions of Sicily and Aragon (as well as Navarre), John would seek another heir. He therefore married Juana Enriquez in 1447. She bore him two children, Ferdinand and Joanna. Charles saw this (correctly) as an attempt to disinherit him from Aragon. John named Ferdinand his successor when Ferdinand was nine years old.
In 1451, Charles rebelled against his father's possession of Navarre and the Navarrese Civil War began, a period of unrest and fighting until Charles was captured in 1452, the same year Ferdinand was born. Charles was released if he promised not to seek the throne of Navarre until after John's death.
All this actually took place while John was still a prince; the current king in Aragon while John was growing up was Alfonso V. Alfonso died in 1458, and Prince John became King John II of Aragon, offering Charles Naples and Sicily. Charles declined, and returned to Navarre in 1459, with a plan to marry Isabella of Castile. That fell through (she later married Ferdinand, of course). Charles died in Barcelona in 1461; the suspicion was poison, administered by his stepmother Juana.
John had several more years in him, however, and we'll wrap him up in a few paragraphs next time.