I used a nursing cover when I'd nurse my daughter in public. It fit over my head loosely (kind of like a thin cotton poncho, I guess) and covered completely. The neck was loose and allowed me to look down and watch my child (and allow her to look up and watch me). From an outside view, all you could see was a bulge underneath and maybe her feet sticking out the side.
I found it to be an awesome compromise. It was light-weight, which was great because I gave birth in August in Louisiana, the material breathed well, and I wasn't forced to hide away like I was doing something disgusting. And I didn't leave myself exposed to possibly offend others who weren't expecting or wanting to see my body.
As she got older and didn't need to nurse as frequently, we'd give her a sippy cup in restaurants. If I was visiting a friend or relative and she needed to nurse, I'd quietly excuse myself and take her into another room. And at home, of course, I'd whip out a boob whenever she'd need one, because my husband and I were both 100% comfortable with it.
I think the key is finding what works for you and being willing to take others' feelings into consideration and compromise.