We're quite fair-skinned (thanks to our Icelandic and Welsh ancestry), and I had many bad burns when I was a kid. I wouldn't wish that on anyone.
However, I hate sunscreen. Really, really hate it. And the only kind I've found that doesn't burn my kids eyes is Neutrogena, which I'd prefer not to use anyway. So it's big bucket hats for us (and I do mean big.. my boys have huge heads) and we try to stay in the shade as much as possible. I don't worry about it too much if we're in just in the yard in the early morning or late afternoon, though.. but if we're out at midday I'm really conscious of making sure they're either in the shade or wearing a hat. I don't worry too much about their arms unless they're doing something that exposes their arms more.. like playing in the sand or something (where their arms are out in front of them, rather than by their sides like if they were walking).
I see lots of kids at the playground wearing at least baseball caps, so I think it's reasonable to expect them to wear a hat for at least the next few years.