Yes, you can damage your eyes from the sun. It is the same as your skin as it accumulates over time. I wear sunglasses that say they have UV protection. They are not any more expensive and very easy to find.
I also try to put sunglasses on my kids so they do not have damage. I also put hats on my kids as much as possible. If I am going to be in the sun for an extended period of time, I wear a hat that covers my face.
Staying in the shade as much as possible is a good way to protect all of your body from the effects of the sun. We need some sun for our body, but be sure to get it in moderation.