Using ESP8266-01 with Arduino uno R3
To confirm that the serial communication is working, connect the ESP to the Arduino D10 and D11 that is used as SoftSerial pins.

+----+
| PC |
+--/-+
   |       +---------+
   \-------| Arduino |
     USB   +--.------+
              | SoftSerial
              |
           +--\----+
           | ESP01 |
           +-------+


And then upload this code to the Arduino. The code is from https://www.arduino.cc/en/Tutorial/SoftwareSerialExample

/*
 Receives from the hardware serial, sends to software serial.
 Receives from software serial, sends to hardware serial.

 The circuit:
 * RX is digital pin 10 (connect to TX of ESP01)
 * TX is digital pin 11 (connect to RX of ESP01)

 Created back in the mists of time
 modified 25 May 2012
 by Tom Igoe
 based on Mikal Hart's example

 This example code is in the public domain.
 */

#include &lt;SoftwareSerial.h&gt;

SoftwareSerial mySerial(10, 11); // RX, TX

void setup() {
  // Open serial communications and wait for port to open:
  Serial.begin(115200);
  while (!Serial) {
    ;
  }

  Serial.println("Goodnight moon!");

  // Set the data rate for the SoftwareSerial port
  mySerial.begin(115200);
  mySerial.println("Hello, world?");
}

void loop() {

  if (mySerial.available()) {
    Serial.write(mySerial.read());
  }
  if (Serial.available()) {
    mySerial.write(Serial.read());
  }
}