Arduino Nano and ESP8266
The Serial Monitor's baud rate must match what you have in 

Serial.begin(9600);


The EDP8266's baud rate is specified in the line 

ESPserial.begin(9600); 


and that is the one you need to change to try different baud rates with the ESP8266. The newer AIThinker firmwares default to 115200 baud, not 9600 baud.

Be warned, though, that SoftwareSerial isn't that great at 115200 baud - you tend to get some data corruption. It is best to change the baud rate to 9600 baud with

AT+CIOBAUD=9600


I'd also recommend investing in one of the ESP-01 programming boards from eBay. Make sure you get one of the ones with a switch on it that switches between programming and UART mode. Something like this one. It makes working with the ESP-01 much simpler - you can directly use your PC to set it up right for your system, and even upgrade the firmware (or program it directly with a sketch from the Arduino IDE if you install the right core) with minimum hassle.