I just assumed that since he really only had his dad for company in his first 6 years, he wouldn't really have much to do, but watch and learn when his father's entertaining. I don't think his dad could just leave Harry in a hotel or ask someone to watch over him.

Foster care is a possiblity and there's the fact that Harry knew what his name sakes are(who's he's named after) so it stands to reason that he picked up a few tricks, trying to not be lonely.

Also, before he went to Chicago and after he left Ebe's, maybe he worked random side jobs that allowed him to make use of his skills because he was a bit wary of using magic after DuMorne.