So Aberforth characterizes his older brother as secretive by nature and nurture. Albus was raised that way by his mother whom aunt Muriel describes as a "proud and very domineering" witch. Probably wasn't easy, attending Hogwarts and enduring all the whispers about his father slaughtering muggles and not being able to defend him, lying about his sister being home-schooled because of her fragile health, fearing that a slip of his tongue would cause Ariana to be taken away. So Albus had to distance himself from his fellow students, he couldn't confide in them. And then the death of his mother...making him the head of the family, responsible for his brother and sister, which also meant he had to sacrifice his career and return to some backwards village Godric's Hollow and keeping his mouth shut.

And when he does confide in someone and makes a friend, his intellectual equal and lover, Gellert Grindelwald, it ends very badly. No wonder after that Dumbledore blames himself and decides not to trust anyone.

The conclusion he arrived wasn't "secrets are bad, my family was wrong for not seeking professional help of the St. Mungo medwizards and failing to provide competent medical care for my sister. Locking her in a house and keeping her state a secret caused the death of too many people." Instead he blames himself for not keeping the secret better and not 'protecting' his sister but being selfish and easily corrupted by his own desires and aspirations.

Probably because of his upbringing and the circumstances of Ariana's death is why later the Headmaster of Hogwarts and the Head of the Order of the Phoenix doesn't confide in anyone; He only gives the essential information to his pawns subordinates, just enough so they could carry out his instructions. The Order members are used to it, and seldom question his judgement.