You asked

Should my child wear sunglasses?

It’s as important to block ultraviolet rays from your child’s eyes as it is to block it from their skin, so buying them a pair of sunglasses is probably a good idea – and often if you let them help to pick the ones they want it will make life easier when trying to get them to wear them. 
You must make sure that when you are buying sunglasses that the product confirms they block out UVA and UVB rays to ensure you are getting the right level of protection. Without this protection the darkened glasses will simply cause the child’s pupil to dilate which will cause even more damaging rays to get at the eye.  The shade of glass is a personal choice, but something of a medium darkness is often the best choice, so that it gives enough protection from the glare without being so dark it would cause the pupil to dilate which allows more UV rays to enter. It is not the colour of the glass that offers the protection.

More questions

Children's eyes are sensitive so it is vital to protect them from the sun's harmful rays.
Your young child can have their eyes tested even if they can't read
Regular eye exams are essential in order to monitor your child's eyesight.
There are some signs to watch out for if you think your child may have a vision problem. 
Should your child get any kind of chemical in her eye you should treat it as an emergency and bring her to the hospital immediately.
You should take your child to their G.P. or to the hospital if you notice any of these symptoms. 
You should seek immediate medical attention if an object seems to be imbedded in your child's eye. 
If your child appears to have something in his eye you should seek immediate medical attention if it is imbedded in the eye. 
As your baby learns to crawl and begins to explore her surroundings you will need to child proof your house to prevent eye injuries.
Babies and young children are always getting up to all sorts of mischief and eye injuries are quite common. It is important to always consult with your G.P. should any eye injuries occur.