I ran the local Staples' Business Machines department when I was in high school, and opened one of the first Tech Centers. Before I could open it, they sent me to a Staples training course, which was taught by a guy who barely knew basics. The entire content of this course was how NOT to fry some customers computer. This is way over-blown
Speaking from personal experience, components are not as sensitive as they make them out to be. As long as you are not stupid (changing parts while powered on, dragging socks across carpet before working on machine) you should be fine.
That being said, my workbench has an anti-static mat on it, and while working there, I am standing/sitting on an anti-static floor mat. When I want to be extra careful I plug the computer chassis into a wall socket using a 3-prong power cable, that I removed the polarized plug, so the cable is just ground.