How do I offer free local delivery but charge shipping everywhere else?

Combine one shipping zone with your normal shipping:

1. Go to Plugins → Shipping Zone, click + Add Zone, name it, for example Local, and click Create. The zone editor opens.
2. Under Regions, click + Add region and add your city (pick Country, State and City). Add more rows if you cover nearby cities.
3. Under Shipping methods, click + Add method, set Type to Free shipping and Label to "Free local delivery".
4. Click Save zone.

Then set the charge for everyone else on Settings → Store Settings → Shipping Setup and click Save Changes. Addresses outside the zone pay that.