Trainers
A neutral running shoe in the kid's correct size. Not a court shoe. Not a basketball shoe.
Replace every 300-500 miles, which for a youth athlete is roughly every season.
Sprint spikes (100-400m and hurdles)
A stiffer plate, six to eight pin holes, lightweight upper.
Buy a half size smaller than the trainer. Sprint spikes should feel close to the foot.
Distance flats / spikes (800m+)
For 800-1600m: a flexible spike with four to six pins.
For 3200m+ on track: a flat or a softer spike.
Jump spikes
Long jump and triple jump: a spike with a heel plate. Different from a sprint spike.
High jump: a high jump spike, sized to the takeoff foot.
Throwing shoe
A flat-soled shoe with a smooth bottom for the throwing circle. No spikes.
Brand and exact model matters less than fit.