The low cost build has been stripped down to the bare minimum in terms of features and expense.

It should be very easy to setup and requires almost no soldering. It requires the following materials which should cost about $40 plus shipping. The files for this build are found in the \Arduino\low_cost\ folder.
Adafruit HUZZAH ESP8266 breakout $10
https://www.adafruit.com/product/2471

USB to TTL Serial Cable - Debug Cable $10
https://www.adafruit.com/products/954

DHT22 temperature-humidity sensor $10
https://www.adafruit.com/product/385

5V 2A (2000mA) switching power supply UL Listed - $8
https://www.adafruit.com/products/276

Female DC Power adapter 2.1mm jack to screw terminal block $2
https://www.adafruit.com/products/368
First you will have to setup your computer so you can upload code to the device. Adafruit has the process documented well here.

For wiring you will need to hook up the following:

for programming and serial output wire TTL serial cable as listed in the adafruit page. this usb connection can also power it while connected to PC.

for standalone you will need to wire VBAT and GND to a 5v power supply such as the one listed above.

The DHT22 should be wired to the 3v power, ground, and pin 2

Before you upload the code to the device you will need to make a few minor source code edits to connect to your wifi network, and your own web client ID. (See the webserver link to the left on how to get your web client id)

Basically all you have to do is rename the public.h file to private.h and edit it. A sample file is provided below:
const char* ssid     = "my_wifiname";
const char* password = "my_password";
const char* host     = "sandsprite.com";
char* APIKEY = "123456";
int   client_id = 0 ;