A natural place to start would be Volume 1 of the so-called GLS project.
Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The Classification of the Finite Simple Groups, Mathematical Surveys and Monographs, Volume 40, Number 1, American Mathematical Society, 1994.
The GLS project is a series of books that is intended to provide a complete proof of the Classification Theorem, modulo a relatively short list of references (such as Aschbacher and Smith's work on quasithin groups) containing crucial pieces of the proof that the GLS authors felt were already satisfactorily written up in a self-contained manner. The GLS project is expected to comprise 12 volumes, of which 8 have already been published. Volume 1 above contains an outline of the entire proof, as it was envisaged in 1994. There have been some changes to the plan since then, but nothing too radical, so Volume 1 still serves as a good starting point. Also you will probably want to read Ron Solomon's progress report in the June/July 2018 issue of the Notices of the AMS, written shortly before Volume 8 was published.