Roy is energetic and all-around. He treats everyone with respect, but will not stand for anybody who wants to hurt his friends or family.
He will first appear two weeks after you trigger Kathy and Owen's marriage ceremony and sequence of rival events that occur before they both decide to get married. Two weeks after their wedding, Roy and Kathy will send you a letter, which you will receive in your mailbox. By reading it, you'll learn that Kathy and Owen have given birth to a baby boy, Roy. They'd like you to come and visit their new child.
Go to the Blacksmith's to visit Roy, Kathy, and Owen. As a newborn baby, you cannot interact with Roy. He'll remain in his crib until he's fully grown. Until this time, you will be unable to make any progress on his friendship points. Until you visit Owen and Kathy at their house, Roy will not appear in game.
(NOTE! He will NOT exist in game if you have both Kathy and Owen unmarried or have married either of the two.)
Roy (Grown Up)
Two weeks after Roy is born and you've visited Kathy and Owen to meet him, you'll get another letter. This time, Kathy and Owen will tell you that they're going to take Roy for pictures at Simon's to celebrate his growth. They want you to meet them there, so you can introduce yourself to Roy.
Go to Simon's Photography Studio any time after getting the second letter, at a time when the store is open. Here, you will see Owen and Kathy, as well as Roy. Roy will introduce himself, and from this point on you can socialize with him. You will be able to interact with him like any other villager, and befriend him. Roy does not get any older in game.
Exploration of the Mine
When you pass through the mine road going towards the blacksmith, you may trigger the event where you see Roy and your child asking each other if they are strong. As expected, both agree and jump in glee. Roy will ask him/her if he could be the leader, your child responds with a yes. Both will leave for the mines and start their adventure.
This event can be triggered with Heath and Lucy as well, and will play out the same. Each child you trigger this event with must be at 2 hearts or more.