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). Note that only the local shipping carrier's tracking information is synced to your store.
Only the local, last-mile carrier's tracking is synced to your store: for example USPS for the United States and DHL for most of Europe. Origin and international transit information is not included, so the tracking your customer sees starts when the package enters the destination country's carrier network.
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.