I recommend a Brachiograph build. It will introduce you to some fundamentals of PWM and inverse kinematics. It is well documented but not cookie-cutter. Using a Raspberry Pi will give you more direct access to running the servos than the microcontroller experience. All the parts are infinitely reusable afterward if you don't want to keep it around.

https://www.youtube.com/watch?v=u4Jh1daCl60

https://www.brachiograph.art/

https://github.com/evildmp/BrachioGraph

  Sample Supply List for $80 budget:
  Pi Zero with header $20: https://www.adafruit.com/product/6008
  Power supply $9: https://www.adafruit.com/product/1995
  SD Card $10: https://www.adafruit.com/product/1294
  Three hobby servos $18: https://www.adafruit.com/product/169
  Breadboard wires $5: https://www.adafruit.com/product/153
  Breadboard $5: https://www.adafruit.com/product/64 
  Glue, popsicle sticks, pen and paper $10