Starting out with robotics without hardware

Hi! I'm a software guy and would like to start out doing some robotics. Before I go out and get a bunch of hardware. I'd like to practice the fundamentals.

I'm most comfortable with C++ and C# and dotnet and am pretty comfortable with game engines like Unity Unreal and Godot.

I've started out modeling a three-joint articulated robot arm that i can control through signals to the individual joints, like controlling a stepper motor.

My goal is to figure out a system where I can declare the shape of a robot like this (armature size, number of joints, offsets etc) to create a virtual model of the robot. I want to be able to send target coordinates and a basis rotation to that model and receive a series of signals back that will move the head of the robot to that 3d coordinate and rotation.

Now, I'm sure there are systems and packages that do all the math for this already, so what tools/libraries do you guys use to do modeling like this?

I want to see if I can simulate it in a game engine, and if that works out maybe ill try it on a toy :D

Thanks!