Would it be safe for an American to go to Serbia?

YES. Belgrade is 100% safe. Oh and you're not alone, there are a lot of American tourists in Belgrade during summer and I love watching them with their surprised looks at how friendly people are here and how casual and laid back everything is. You can contact the American emby in Belgrade and they will tell you about the amazing nightlife, food, history and anything else you want. Nobody will giveyou dirty looks. They often ask if you are American if they hear you speak English but then they're totally into you like "omg i have a friend in america etc" or something like that. You won't get any weird looks, as long as you behave normally. If you are interested in yugoslavian history, you might want to go to Bosnia as well. Beautiful country, great people, great food.