Connecting esp8266 using Arduino
There are a number of issues with the tutorial you've listed there.

Firstly, a voltage divider is very highly recommended as the ESP8266 cannot handle more than 3.3V, however, the connections for the voltage divider in the tutorial you linked to are the wrong way round (as you suspected).

Secondly, you cannot power the ESP8266 directly from the Arduino, as the power requirements are higher than the Arduino's on board 3.3V regulator. A separate power supply or regulator is recommended. This might be cause of it getting hot (the onboard regulator maybe - but I can't be sure as there is not enough information from your initial post as to exactly what is getting hot).

There are 2 LED's on the ESP8266 shield (the one used in the tutorial you've linked to) one is power and the blue one is wifi communication.

The other potential issue with the tutorial you've linked to is that they have used Arduino pins 0 and 1 to connect to the ESP8266 - while this will work, the problem is that the Arduino uses pins 0 and 1 to communicate over USB with a computer, which means that in in that configuration you cannot use Serial monitor to send/receive data.