As someone who's built both BattleBots and Professional Robotics for work, BattleBots is a great way to get out of equations and hands on fabrication, manufacturing, testing, and scrappiness that is so hard to reach in mechanical and electrical engineering. And unlike FIRST or Lego robots, it's much more open ended and "guardrails off" engineering, which I found really freeing from the tyranny of academic-style competition robotics. You can still incorporate all the sensors and algorithm-stuff (many folks build their own motor controllers like "brushless-rage" or have sensors like Chomp), but if you just love seeing things move and love mechanical design, it's a great thing.
For BattleBots in particular, the easiest way to get into it is to find some guides online for a simple bot[1] with DC motors and a 3D printed body, and just enter it into a local combat robot competition! You'll learn the basics of a motor, speed controller, selecting wheels and other interfaces, as well as designing a chassis and fabricating it. At a competition you get the thrill of the fight, and afterwards you can sweep your robot scraps into a dustpan, make friends with other bot builders and go from there.
[1] A quick search on instructs Les and I found this, though there are many more great robot tutorials: https://www.instructables.com/Naked-Singularity-Beetleweight... . Here is one that overviews all the basic steps in a BattleBots https://www.instructables.com/How-to-design-and-build-a-comb...