First get a decent power supply. The 9 volt battery (and onboard regulator) don't have the ability to supply the power requirements of the ESP.
The ESP-01 uses a serial interface so you either connect it to the serial pins (D0,D1) and disconnect the USB cable or use software serial on two different pins.
Using an ESP8266 board like the NodeMCU or ESP32 and a DS18B20 temperature sensor would be easier than your current setup with adding on a ESP8266-01. No Nano required and you have WiFi.
It is not just programming 2 processors to talk to each other. If the Nano is a 5V unit you have to contend with levels shifting the Nano 5V TX to the ESP 3.3V RX and providing adequate 3.3V to the ESP (Nano 3.3V will not cut it).
I gather the blue board you cite is a gas sensor. This requires 5 V and draws significant current so you need a 5 V supply to power it. The Nano is not a 5 V power supply so you have a problem there.
The ESP-01 does not have the analog input connected, otherwise it would do the job. Looks like a WeMOS D1 Mini is the answer, when you find a suitable 5 V power supply.
NodeMCU is an unnecessarily more expensive version of the WeMOS D1 Mini; no advantage.
If only a temperature measurement was wanted, a DS18B20 could be directly used with an ESP-01 and powered by two "AA" cells, but this appears to be using a gas sensor.