Communicate with ESP8266 RX/TX
Your analysis is indeed correct, you can communicate between Arduino and ESP using the serial protocol. 

Electrically speaking you must ensure the RX and TX pins are crossed: 

Arduino RX goes into ESP TX
Arduino TX goes into ESP RX

Also, the two devices MUST have a common ground, the GND pins of the Arduino and the ESP8266 must be tied together.

You don't need level shifters because, even if the ESP8266 runs at 3.3V, its IO ports are 5V tolerant.

Also, as you already pointed out, the two devices must set the same speed (aka BAUD rate) in order to comprehend each other.

In case you decide to transmit binary data please also consider the two devices use a different architecture: Arduino is 8bit, ESP8266 is 32bit, meaning they have different word size (the minimum allocation unit size).

I had to face the word size issue when I first tried to exchange a struct between an Arduino and a RaspberryPI (also 32bit)...