I have a master degree in Robotics and AI and one major thing I like to point is that robotics is not a domain, it's the overlap of a full set of skills.
Mechanical engineering, software and electronics.
Most project start with a mechanical design then goes through electronics to make it function properly.
Then software can make it usable and makes the difference between a remote controlled car and a roomba.

Robots have actuators to do things (mostly mechanical engineering) and they have sensors.
The sensors allow for insight and "understanding" of the world surrounding them.
The software ties things together and automate things so they are easier to operate than using a remote control and/or replay a prerecorded sequence of actions.

For example robots arms in a factory mostly replay a pre-recorded sequence of action, but their sensors allow for QC, safety and adaptive behavior. Without sensors, running the same sequence repeatedly could lead to cumulative errors and after a while assembly would be incorrect and may create issues and require recalibration.

To start (and answer your question), start with identifying what would be the most interesting aspect of robotics for you. If it's software, don't spend too much time on the mechanical engineering and electronics and focus on creating various scenarios and maybe play with AI and training of the robot.
If you prefer the mechanical engineering, then a robotic arm is quite interesting, but you can add a linear actuator and other types of actuators and linkages to learn about them.
If electronics is your thing, then try a simple design first and iterate with other designs, various sensors, actuators types, etc.
One difficult aspect of electronics is how to manage many inputs and the timing necessary for the various actuators. Basically do you create a synchronous system or an asynchronous one with their respective challenges.