I'm starting to learn set theory. I've already learned transfinite induction and I started to wonder if we could do the same on a well-ordered proper class, like the class of all ordinal numbers. It seems intuitive to me that we should be able to prove statements about all ordinal numbers by induction. But I'm not sure what theory I need to do this. I don't know anything about set theories other than ZFC, and I still don't really know much about ZFC, so I'm not sure what is and what isn't allowed in ZFC regarding proper classes. For example, I don't know if it's possible to say in ZFC that the class of all ordinal numbers exists. I think it might not be possible. But then maybe I don't need to talk about this class to perform induction over it. Maybe I just need to know what an ordinal number is, and this I know to be sayable.
Could you help me understand this? Please, if there is the slightest chance I won't understand a term or a notion, assume that I won't. I'm new to set theory.