WRL-1368 (ESP8266) wifi module briefly flashes blue LED once when powered up, then nothing happens
It turns out that newer variations of the module don't have a red LED, to limit power consumption, so the single blue flash when it powered on, was expected behavior

I got it to work by:


Loading an empty program into the Arduino UNO
connecting TX on the Arduino to TX on the ESP8266, and RX to RX. I build a voltage divider circuit to drop the RX from 5V on the Arduino to 3.3V on the WRL-1368.
Connecting 3.3V to VCC and "Enable" on the ESP8266.
Connecting ground to the ESP8266 and to the reset pin on the Aruduino. I let the reset pin, and other 2 pins on the ESP8266 float.


After this, I was able to issue AT commands.

I used a combination of these two tutorials:

https://www.geekstips.com/esp8266-arduino-tutorial-iot-code-example/
http://www.teomaragakis.com/hardware/electronics/how-to-connect-an-esp8266-to-an-arduino-uno/

EDIT:
Based on Chris Stratton’s comment below, I’lll be reintroducing the voltage divider.