Setup for Arduino and ESP2866 for WiFi/Internet Control

I'm doing background research to make sure I have the necessary parts to setup my project. The general idea for the project is to develop an Android app to control my Arduino over WiFi (doing things like switching LEDs or moving servos).

So far the components I have purchased are:
-Arduino MEGA2560
-ESP01s board (ESP2866 chip) from here
-Electronics kit (resistors, potentiometers, wires, LEDs, etc)

So I've done a lot of reading and I get the general idea of using two serial ports on the arduino, one to communicate with computer, one to communicate with the ESP board (I'm not interested in programming the ESP as a standalone because I need to use sensors and motors with the Arduino).

My biggest question is how to setup the ESP/Arduino system to be controlled over WiFi. I see some projects mentioning the use of an FTDI board (I have no idea what this is or how it should be implemented), while others show that the ESP and Arduino are enough to control over WiFi.

Also, how do I setup WiFi communication with the ESP? I see some projects mention that the board can be used itself as a WiFi server...I also have my own domain, if it is possible to use that to communicate in anyway?

I am competent with the hardware and circuit setup, but lack understanding of the communications, one of the reasons I am interested in this project. Thanks!