The thing about the Classification "paper" is that it is not one paper, it is a lot of papers. Like, more than a thousand. You can find multi-volume compilations of these papers in some academic libraries but really there is probably no worse way to learn about the classification than trying to read those.
To be clear, the papers weren't written after the fact, when everybody already had a good understanding of the theory and where it was going. They were written during the decades long process of discoveries, without any of the authors knowing that there even would be a full classification. The result is that they are very meandering and technically complicated, often excessively so. There has been a movement in recent years to simplify the classification theorem, now that we have a much better picture of the field, but it isn't done yet, and to my knowledge they have not released any intermediate volumes lately (EDIT: I'm wrong). In fact, there probably isn't anyone in the world who has read every part of the original theorem fully. It is just too big and too messy.
If what you want is to understand the details of the results of the classification, (i.e. the groups,) then, after learning the fundamentals of finite group theory, the direction to go is to read a book about finite simple groups. There is a good one by Robert Wilson. You can find reasonably good summaries online if you are only looking for a broad sub-research level understanding. This is just not a topic where you want to go directly to the primary source material.
