A Mega (Atmega2560 based board) can control up to 48 servos. Plan for each servo to need about an amp.
Stepper motors needs a stepper motor controller to create the higher current pulses that make the motor stop. Each stepper motor controller will usually need a Direction and a Step pin at a minimum. You can check then out at Pololu.com
Stepper motors will also need an amp or more, depending on the motor.
LEDs, those just need an IO pin and a current limit resistor for the typical 5mm single color LED.
Same for a buzzer.
Electromagnetic sensor, is that a reed relay that closes when a magnet passes it? Please clarify. One input needed for each if so.
Screen, find one with an I2C interface, only 2 pins are needed. There are 2 pins on the Mega dedicated for I2C (SCL and SDA). If you go with a largeer screen, then an SPI interface will be faster. That will connect to 4 dedicated pins, SCK, MOSI, MISO, and SS.
A Mega has 70 IO digital IO pins available; 16 of them can also do analog input. Doesn't sound like you need analog inputs tho.
Sparkfun.com and Adafruit.com are also good sources.
Power supply: you may need to acquire several 10A power supplies to power the various motors.
All Grounds will need to connect together and also connect to the Arduino Ground to have a common reference.
One 10A motor might supply 10 servos for example, with 10 control signals coming from the Mega.
MPJA.com has a lot of supplies available, in different voltage and current ratings. Hard to recommend past that without knowing more about your servos and stepper motors.