Helmet
Purchase a CPSC-certified helmet, preferably with MIPS technology that protects your head and life in a crash.
Lock
Always buy a secure lock, either a chain lock or a U-lock. Do not buy cable locks. They are cheap and flimsy.
Bottle
Food and water is the fuel of a bicycle, as is gas is the fuel of a car. Buy a bottle that has a squeeze-spout, like a sports bottle. A common brand is CamelBak, which also makes insulated bike bottles. Please clean your bottle before using it.
Tires
Tires are one of the most important things of a bike. The type of tire decides the capability of your bike. If you like to go offroad, you should buy gravel tires. They're good on the streets and trails. If you ride on the streets, you should buy road tires. Make sure your tires fit, and always buy new inter tubes with new tires.
Bike
A bicycle from any big-box retailer will work, but make sure that it's assembled correctly!!! Only buy it if your child isn't going to use it many times each week, the parts are usually low-quality and incorrectly assembled. In fact, if your child cycles a lot, you should buy from a bike shop instead of a big-box bike.