I’m only on season one, currently, but I really want to know if he does ever use his power without it breaking his bones.
Yes, he was able to. In the later seasons, he kept check of his power and only uses less than a hundred percent to avoid injury. To be more precise,
If Izuku does not keep One For All below his current limit of 20%, his body can be immensely damaged by the force of his movements. (Wiki)
During the Internship Arc, however,
he was able to use 100% of his power but with the aid of Eri. As seen in Chapters 157 onwards, this was possible since Eri has the ability to rewind the state of the body of those she touches. Izuku's Quirk destroys his body while Eri rewinds it to a previous state or 'heals' it simultaneously and thus, making him maintain 100% power during this particular fight without breaking his bones or damaging his body.
In season 2 episode 4-5 (I think), Deku uses his quirk One For All for the first time without breaking his hand, but he did say that his hand still hurt but it wasn’t broken, so he was fine. I think that was the first time he uses his quirk without breaking his bone/bones.
He won’t have any bones broken as long as he keeps his quirk to 20 percent and below.