I got one of those from Aliexpress(looks like you were way overcharged!) and it came with the DOIT firmware. I was able to follow the instructions at that gitbooks link and access the configuration web page. It sounds like yours doesn't have the DOIT firmware on it, no big loss. The DOIT firmware might be ok, I don't know, but any example code or library you find is written for the espressif AT firmware so it was useless to me. I updated to the latest AT firmware right away using an FTDI breakout following the standard process for manually installing the AT firmware on an ESP8266. I can post instructions if you want.
Once I had the standard AT firmware on there I was able to successfully use it with the WiFiESP library(GitHub - bportaluri/WiFiEsp: Arduino WiFi library for ESP8266 modules) and have had no problems.
I wouldn't bother using the thing standalone, there are much nicer standalone ESP8266 boards for less than $3, I like the WeMos D1 Mini for that purpose.
The trick with these shields is they are connected to pins, 0 and 1. On Uno or Mega that's Serial. This means that when you upload a sketch to your Arduino board you need to turn off the switch on the shield so it doesn't interfere with the upload. It also means you can't use Serial for the typical purpose of debug output. You're working blind with these things. You need to make sure your code isn't sending anything except AT commands on Serial. The author of the WiFiESP library made the genius move of leaving debug output to Serial on by default so you need to go and disable that to make the library work with an ESP8266 connected to Serial(Logging disabled by default by per1234 · Pull Request #46 · bportaluri/WiFiEsp · GitHub). You also need to use the right baud rate. The default for the AT firmware is 115200.