Hi Blair,
To keep this project simple, you could start with an Arduino Uno:https://core-electronics.com.au/uno-r3.html
and use this servo shield:
https://core-electronics.com.au/adafruit-16-channel-12-bit-pwm-servo-shield-i2c-interface.html
There are great instructions on how to program the Arduino and servo shield here:
We have some great tutorials and workshops on programming the Arduino:
Hopefully that will get you started!
The 4AA battery pack should work fine for this many servos.
