Arduino Zero ESP8266 AT Command troubles

Using Arduino Networking, Protocols, and Devices
2

That information is only correct for boards without native USB, such as Uno, Nano, Mega. The Zero has native USB so that won't work.

You need to run a sketch on the Zero that will bridge communication between Serial (the communication with Serial Monitor via the USB cable) and Serial1 (the communication with the ESP8266. You will find such a sketch at File > Examples > 04.Communication > SerialPassthrough. You only need to change the baudrate settings in that sketch to 115200 instead of 9600:

  Serial.begin(115200);
  Serial1.begin(115200);

You need to connect the Zero's pin 0 (RX) to the TX pin on the ESP8266. You need to connect the Zero's pin 1 (TX) to the RX pin on the ESP8266. The reason for the RX-TX, TX-RX connections is that TX stands for "transmit" and RX stands for "receive".