You definitely don't need an engineering degree, but it wouldn't hurt.
The depth of knowledge you need will be determined to some extent by what games you want/have and what you're doing with them. E.g., if you bought that Soul Calibur and it worked fine, there's not really a whole lot of "new" info you'd need. Common sense dictates that you don't go poking around at high voltage stuff (monitor, power, yada) and so forth, if it's not broken, don't fix it. There's nothing to connecting a mobo, you just plug it into the edge connector on the harness and away you go.
On the other hand, if your first machine is a dirt cheap Galaxian with no power or a monitor in bad shape, that's something else. Generally, you'll want to read up on cap kits, basic wiring, monitor discharging, etc. which should be easy to find at the links in the sticky post atop this forum. But if you have more specific questions on a particular game / problem, the gang here will do their best to help you out.