So you should add a voltage divider on the ESP's RX pin. Arduino RX -> 1K -> ESP RX -> 2K -> GND, to prevent 5v on the pin. Those pins are not 5v tolerant !! Your ESP will probably break at some point if you don't add the voltage divider.
And i suppose you have not uploaded anything to the Mega that uses Serial.
You can communicate between the ESP & the Mega by crossing the TX/RX lines (with a voltage divider on the ESP's RX pin) using any of the Mega's Serial ports. Normally i would suggest Serial 1, 2 or 3, since Serial0 is also used by the USB port.
Then you can setup a communication between the 2 boards.
Have a look at Serial Input Basics for reliable methods of reception (keep in mind that you have to send the data in such a way that you will receive it the way you want)
The whole thing that refers to the LCD & servos should be uploaded to the Mega, since i guess you have connected those parts to that board.