The (only) correct grammar here is Я учился во многих школах; while you shouldn't worry about "studying in many schools at the same time" (you'd need to explicitly say сразу во многих or сразу в нескольких), there is a different kind of vague problem about this wording, in that it sounds more like "I studied in many of the schools". Russian doesn't have articles, of course, but some random distinctions based on definitiveness turn up every now and then; многие sort of implies that more information about those particular schools is forthcoming ("in many of the schools which...") or has already been given.
Я учился в нескольких школах doesn't have this problem; I don't know how many schools we're talking about here; I went to five, but it's still something I would say, i.e. I don't feel "several" is an understatement. If you insist on "many", though, it's probably better to reword as Я сменил много школ. (Another option, theoretically, would be Я учился во множестве школ, but that would most likely be an overstatement unless we're talking about a really extraordinary number of schools, maybe about ten or more.)