Trainers
A neutral running shoe sized correctly. Take to a running store for a gait check at the start of the season.
A second pair on rotation extends the life of both.
Racing flats
Optional through age 12. Worth it at age 13+.
Lighter and stiffer than trainers. Not for daily training.
Spikes (cross country)
Cross country spikes have aggressive lugs in addition to pins.
Different from track spikes. Track spikes do not bite on grass and mud.
Cold weather
Tights, a long-sleeve base layer, gloves, a beanie. Cross country season runs into November in most regions.
Layers come off mid-race in some warm-late-fall regions. Practice with the layers the kid will race in.