What sort of education/degree(s) do I have to receive/earn to begin a career in human genetic engineering?

My guess would be biomedical engineering, or possibly just biology. The best way to find out - find a job posting that's exactly what you're looking for, and see what degree they call out in the qualifications.

Well, clearly you'll need at least one advanced degree in Mad Science (most Mad Scientists prefer the Doctorate, but some have done just fine with a Mad Science Master's). That's not the hard part, though. The really hard (and often cripplingly expensive) part is establishing your research center (or Secret Lair). Ideally, it would be an ancient but structurally sound mountain-top castle, but some ambitious Mad Scientists have opted for the more impressive Underground (or even Undervolcano!) Complex. Make sure that it's on an established shipping route - special handling costs can really eat up a budget, especially if transport involves hand-carrying across poorly-maintained rope bridges spanning dizzyingly deep mountain gorges with fuming torrents appallingly far below. Anyway, that's off in the future; for now, the question is where to get the degree. Most universities are too hide-bound to really get into this field, but New Jersey can always be depended upon for this sort of thing, so the State University offers both MS and PhD in Mad Science, and even EdDs if you want to set yourself up as the Dean of your very own Mad Science U. Harvard offers all the courses you'd need, although at exorbitant cost, and also helpfully includes in all their courses (at no extra cost) a section on "data fudging, manipulation and outright prevarication for fun and profit", as well as "plagiarism: intellectual theft or merely sensible outsourcing?". I believe Transylvania University is considering opening up a School of Mad Science, but so far that's merely rumor (although, of course, that's what you would expect it to be). Good luck!