Gi (uniform)
Karate and taekwondo: lightweight, sized by height in inches.
BJJ: heavier weave, sized 0-A6. The school will tell you what they require.
Two gis is the right number. Practices fill the laundry hamper fast.
Belt
Comes from the school. Length should wrap twice around the waist with about a foot of tail on each side.
Mouthguard and protection
Required for sparring in most styles.
Headgear, chest protectors, shin guards: usually sized by age and provided by the school for the first six months.