This is just the first step to make a cheap Arduino WiFi shield using the ESP8266 module. With few more electronic components and the WiFiEsp library you can build it for less that 10 USD.
Step 1: Components
Here are the components you need
Arduino board - In this example I'm using an Arduino Uno board but I personally uprefer using an Arduino Mega because it has more memory and has a second serial port to communicate with the ESP module.
ESP-01 - This is the smallest and cheaper type of ESP8266.
AMS1117 5V to 3.3V Power Supply - Arduino boards are typically powered at 5V while ESP8266 needs a 3.3V power source. The Arduino 3.3V output pin cannot provide the power needed by the ESP (up to 250mA). This can be solved using a 3.3V voltage regulator like the LM1117/LD1117 or AMS1117.
Breadboard - A 170 holes mini breadboard is enough but you can use a bigger one if you need.
10 uF Capacitor - A small electrolytic capacitor is needed to stabilize the voltage regulator. Any 10-100 uF should be ok.
2 resistors (1K and 2.2K) - A simple voltage divider is needed to shift down the 5V output of the Arduino TX pin. You can use other resistors as long as they are in a 1/2 ratio approximately.