Thanks for the question, and congrats on the new iPhone 5s. The following article will help to troubleshoot Touch ID:
Troubleshooting Touch ID
1. Ensure that your fingers and the Home button are clean and dry.*
Note: Cover the Home button completely. Don't tap too quickly, don't press down hard, and don't move your finger while Touch ID is scanning.
2. Tap Settings > General > Passcode & Fingerprint > Fingerprints and verify:
1. Passcode Unlock or iTunes & App Store is enabled.
2. One or more fingerprints have been enrolled.
3. If one finger isn't working, tap Edit in Settings > General > Passcode & Fingerprint > Fingerprints > Edit to delete and then reenroll the finger.
4. If that finger still isn't working, try enrolling a different finger.
5. If you are unable to enroll any of your fingers, take your iPhone 5s to an Apple Retail Store, Apple Authorized Service Provider, or contact AppleCare for further assistance.
If you are unable to complete Touch ID registration in iOS Setup Assistant, restart your iPhone 5s.
* Moisture, lotions, sweat, oils, cuts, or dry skin may affect fingerprint recognition. Certain activities can also temporarily affect fingerprint recognition, including exercising, showering, swimming, cooking, or other conditions or changes that affect your fingerprint.
Enjoy your new iPhone!