Connecting and using ESP8266 wifi My ESP8266 has stopped working.
The red power light comes on and the blue TX light is on solid.

Here's what I did.

Connecting

ESP8266  -> Arduino nano

VCC, CH_PD, RST  -> 5V (external supply)

TX -> TX

RX -> RX

I could send AT commands via the serial monitor.

Then I connected

RX -> 11

TX -> 10

and set up software serial intending to repeat the experiment.
I was getting garbage characters in the response, and found that I had to esp8266.println("AT+IPR=9600"); to reduce the baud rate.

Then the blue TX light cam on permanently and it's stopped working at all.
Serial devices should be connected RX to TX and TX to RX.

When you connected the RX to RX you connected the esp8266 to USB chip on Nano. Because the USB chip is connected to Atmega on Nano RX to TX, so if you connect the esp8266 to the same line, the it is connected RX to TX to USB.

On this picture esp8266 is connected to Atmega and not accessible from USB:



Communication from Serial Monitor over USB to esp8266 on pins 10 and 11 of the Atmega could work only with SerialPassthrough sketch resending data between Serial and SoftwareSerial on pins 10 and 11.