Henry VII and Elizabeth were both grief-stricken, of course, at the loss of their eldest child. Within months Elizabeth was pregnant again -- possibly, it has been suggested, in an attempt to bring forth another son. Henry had spent a good portion of the last 17 years blocking plots to overthrow him and eliminating rivals to the throne. He was very much aware of the importance of securing the Tudor dynasty with male heirs -- an attitude he imparted to his surviving son, the future King Henry VIII. Unfortunately, the pregnancy cost Elizabeth her life.
Because Arthur was expected to take the throne and the spotlight was on him, relatively little was recorded about young Henry's childhood. He had titles and offices bestowed on him when he was still a toddler. His education may have been as strenuous as his brother's, but it's not known whether he received the same quality instruction. It has been suggested that Henry VII had intended his second son for a career in the Church, although there is no evidence of this. However, Henry would prove to be a devout Catholic.
Erasmus had taken the opportunity to meet the prince when Henry was only eight, and had been impressed by his grace and poise. Henry was ten when his brother married, and he served a prominent role by escorting Catherine to the cathedral and leading her out after the wedding. During the festivities that followed, he was notably active, dancing with his sister and making a good impression on his elders.
Arthur's death changed Henry's fortune; he inherited his brother's titles: Duke of Cornwall, Earl of Chester, and, of course, Prince of Wales. But his father's fear of losing his last heir led to serious curtailment of the boy's activities. He was given no responsibilities and kept under close supervision. The ebullient Henry, who would later become renowned for his energy and athletic prowess, must have chafed at these restrictions.
Henry also appears to have inherited his brother's wife, though this was not at all a straightforward matter.