"Fit into" is usually about physically fitting inside something, i.e. "Do you think I can fit into clothes that small?", "This jigsaw piece fits into that part of the puzzle". "Fit in" is more social, i.e. "I don't fit in with those people", "I hope I will fit in at my new job". "Fit on" again is physical, but on top of something rather than inside it. i.e. "A few more books could fit on that shelf", "It's hard to fit such a small hat on my head". "Fit" on its own can sometimes be used in place of "fit into" or "fit onto", and has a few other uses too. For instance, "they aren't a good fit for each other", "those clothes are ok, but they could fit better".

Try "Cambridge dictionary online" under "fit"