Once an order ships you receive the tracking number by email, and it syncs to your Shopify store automatically, so your customers see updates under your brand (set up Shopify email notifications to keep them posted).
No movement right after fulfillment is normal: the tracking number is generated immediately, but carriers show nothing until the package is physically scanned into their network, typically within 24–48 hours. If tracking hasn't updated after 48 hours, contact support with your order number.