I have seen a lot of negative reviews and even boycotts of some Elsevier journals. I do not want to jeopardize the future of my academic career by publishing in illegitimate journals. Am I correct in assuming that the journals on Elsevier are considered "legit" but just "very pricey," and this is the source of the negative reviews? Would it be safe to publish on any journal of Elsevier?
They're generally legitimate. The negative reviews and boycotts aren't about the quality of the journal, but are because of Elsevier's alleged high prices (see The Cost of Knowledge). Elsevier publishes some of the best journals in some fields, e.g. The Lancet and Cell.
Having said that Elsevier gives their editorial boards some leeway on what to do with their journals, and to my knowledge up until recently that included not conducting peer review if they don't think it's necessary or appropriate. This journal Medical Hypotheses was an example. That journal published a paper denying that AIDS is caused by HIV, there was a backlash, and Elsevier replaced the editor-in-chief. Although you should be fine in general if you publish with Elsevier, to be sure, you'll have to research the journal in question.