Those little ear plugs are always an expedient; and, after a few hours, they can get damned uncomfortable to wear, too. Sometimes I'll wear them underneath my ear muffs, though, like when I'm shooting magnum cartridges or 50 caliber. What you, normally, want is any ear muff type hearing protector with a decibel rating at or above 27dB.
Yes, the safety glasses you got at Home Depot should be sufficient; however, my own safety glasses have always had side shields installed on them, too. As a matter of fact, I don't skimp on eye protection. I went to my optometrist and had him make me up a pair of Photo-Gray safety glasses with heavy glass lens and industrial side shields. (Buy them once; keep them forever; but, this is not something that you absolutely must do. However, I did!)