When Forrest first hears that there's a Forrest Jr. running around, his first question is whether he's "like me." Now, usually a dad wants to hear that his son is like him—but not in this case. Forrest is super relieved to hear Jenny say that Forrest Jr. isn't like him at all—that he's "very smart. He's one of the smartest in his class."
Once Jenny dies, Forrest takes on the role of being Forrest Jr.'s only parent. On his first day of school, we see the boy go through the same routine that Forrest did. The last thing Forrest tells his son in the movie is that he loves him, to which Forrest Jr. answers, "I love you, too, Daddy."
Looks like they're going to do just fine together.