You have received good advice so far. Just a few of my own thoughts on the matter.
1. Don't just settle for a "snorkel set" of fins, mask and snorkel sealed in plastic. The only advantage is colour-coordination and it's unlikely otherwise that everything will match your child's dimensions. Fit matters for snorkelling to be an enjoyable experience.
2. Plastic-bladed open-heel fins may have street-cred with young people who want their gear to resemble what adults wear, but such fins are normally worn with neoprene booties to protect feet, even by grown-ups, meaning additional expense. Plastic blades can also be somewhat stiff for growing feet. Decades ago, in Eastern Europe, children and young people would often enter finswimming races using age-inappropriate fins with long stiff blades, sustaining skeleto-muscular damage in the process after prolonged use. Snorkel rental outlets in Hawaii offer all-rubber workout-type full-foot fins for snorkelling which are comfortable and easy on the feet and economically priced. As a lifelong snorkeller I've always swum with such fins and they are so comfortable I almost forget I am wearing them. Whatever kind of fins you eventually choose, make sure that they are a snug and comfortable fit for your child, who may otherwise suffer from cramps or blistering.
3. The importance of mask fit has already been mentioned. An ill-fitting leaky mask is miserable for any snorkeller and even more so for young people. The traditional way to check that a mask matches facial dimensions is to place the inside of the mask on the face without fastening the strap and to breathe in through the nose to make a partial vacuum. If the mask stays in place without the strap, the mask fits, at least theoretically. Wearing the mask with a snorkel attached in the water is the only way to be sure that the mask is a perfect fit.
4. Choosing an age-appropriate snorkel is also crucial. Unfortunately, snorkelling gear retailers don't make purchasing one easy. There are national and international standards for snorkels, all of them focusing on the length and the inner diameter or capacity of the barrel:
Note how these dimensions have changed over time as science and technology have progressed. The latest European Standard EN 1972, dated 2015 and entitled "Diving equipment. Snorkels. Requirements and test methods", sets a maximum length of 350 millimetres and a maximum inner volume of 150 cubic centimetres for breathing tubes used by people with small lung capacities. That includes small children, who may not have the pulmonary strength to breathe easily when using an adult-sized snorkel head downwards on the surface, or to blow water out of the barrel when surfacing after a dive.
