Fascinating! I want to get into this type of stuff. But I have no idea where to start, I just have just a CS degree and 3 years experience as a developer.
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.
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
Arduinos and hobby servos. No, neither of them are "industrial grade" and yeah, you'll reach their limits pretty quickly, but building a physical thing that does stuff is (in my experience) a huge motivator.
Or if you're already all over the basics, figure out what kind of stuff you want to build and then try and build it. :)
To begin with you can get a cheap robotic kit from Amazon, there are many of them, and put it together. That's probably the easiest and fastest way. From here you can read more about servos and controllers, modify its mechanics and software.
https://www.youtube.com/watch?v=u4Jh1daCl60
https://www.brachiograph.art/
https://github.com/evildmp/BrachioGraph