To connect the ESC to the FC please match the wiring diagram of your ESC to the wiring diagram pictured below. You will see that the connector version has 2 power and 2 grounds going into the FC. This redundancy helps prevents damage to the FC if either the positive or ground gets disconnected. The extra ground is the most important. If your ESC does not have 2 ground wires coming out, then you can wire directly from the FC GROUND to the ESC Ground pad that you connect your battery lead to. On the solder version there are only pads, so to get this redundancy please at least add an extra ground wire from any ground pad on the FC to a ground on the ESC.
There is a pad on the ESC for current and telemetry. If your ESC has an output labeled CUR or current then connect to current on the flight controller. If your ESC has telemetry output or TX output, then connect to the Telemetry pad on the FC.