If I use Shipping Zones and a courier, which one sets the shipping price?

At checkout, the price comes from your zones or your manual shipping, not from the courier:

1. If the Shipping Zone plugin is active and the customer's address matches a zone, they pick one of that zone's methods and pay its cost.
2. Otherwise, the Manual Shipping settings on Shipping Setup apply: a flat Overall Product charge or Per Product costs, with free shipping above your minimum if enabled.

A connected carrier is used after the order is placed. In the order's Fulfilment & Tracking panel, pick the carrier, then click Create shipment via it (or type the AWB and click Mark Shipped), and use Track to follow the parcel.

So set your zone or manual prices to roughly cover what the courier charges you. The carrier settings mention live rates, but the storefront checkout doesn't use carrier rates at the moment.