Your baby can start wearing sunscreen at about six months age. Prior to six months of age, we recommend you really keep your baby in the shade.
Car seat carriers now have sunshades installed on them, and you can even go to your local baby store and buy these giant sunshades that strap on with Velcro, and you can strap them onto your car seat or you can strap them onto your baby's stroller to make sure that your baby's whole body is covered.
That being said, if you know that your baby is going to be exposed to the sun and you can't find a shady spot, you could put the sunscreen on the baby's arms and feet. If they're younger than six months, there's no study that shows that you can't do it, but we can't recommend it.
I always recommend that the baby wears a hat. They even sell some baby sun glasses if your baby's eyes are sensitive to the sun; but really what's imperative is a hat. You can just have the baby dressed in clothes or if you are going to be at the beach you might want to buy the clothes that come with the SPF already embedded in it to offer your baby more protection. Really the best thing is shade if you're going to go out and know that you're going to be in a sunny area, just come well prepared.
I don't recommend putting sunscreen on an infant's face because if they sweat and it bleeds into their eyes, they can a corneal burn actually, and the babies have no way of communicating that their eyes are burning. That's why I really recommend keeping a hat on a baby all the time and ideally until six months of age.