[jmptable]

I would recommend starting with simulators! Industrial robot arms are expensive and the cheap arms designed for hobbyists aren't very useful because they lack payload capacity, reach, and degrees of freedom. But there are good simulators out there which let you build software or test applications with robot arms without having any real hardware. You can even have real-time physics-based interactions with the simulated world. It can feel like a video game.

My favorite for messing around is CoppeliaSim. It is relatively easy for beginners (you can just drag robots from the library into your sim scene and they will start doing things) but also powerful. It has a free education version but unfortunately it's commercial software. There's also Webots which "feels" similar but is open source. I found it a little harder to get going in but still very capable.

On the harder but more rewarding end of the scale there's the Robot Operating System (ROS) + Gazebo. ROS is used for many real robots both in academia (see the ETH Zurich work on ANYmal) and in industry (see Ghost Robotics). Gazebo is a simulator that is designed to work in ROS. You can build a control system on top of ROS and watch your virtual robot solve problems in Gazebo, and then you can flip the switch and run it with a real robot.

If you do decide to buy a robot arm I would recommend myCobot from Elephant Robotics. It's cheap (~$600), made of plastic, weak, etc. but has 6 degrees of freedom and enough capability to do real projects. If you play with something like that and get obsessed then I recommend taking a look at robots on eBay. Kind of counter-intuitively used giant heavy metal industrial robot arms are often cheaper (~$20k or less) than the "cobots" that are small and safe ($50k or more). That's because so many of the giant ones are out there in the world and are reaching the end of their normal lives. Sometimes owners are willing to pay to have them removed because it's such a hassle to deal with them. The big ones have to be bolted to the floor (after a structural engineer gives the ok on the dynamic loading on the building they're being put in) and you need industrial power. Also they often have archaic control systems running Windows (e.g. Kuka KRC) that have to be coaxed back to life with sparse information. If you go that route then the robot-forum site will be your lifeline.

On the development tool side I highly recommend finding a robot that already has ROS support. E.g. the ROS Industrial Kuka Experimental package for Kuka arms. The actual underlying interfaces/APIs (e.g. RSI or the fast research interface for Kuka arms) can be a massive pain to deal with. Newer robots are much nicer to deal with. There are a lot of cheap arms coming out of China that come with real-time ROS compatibility out of the box. Like the Z1 from Unitree, which I'm very excited about. Universal Robots bots are also well-supported in ROS and the underlying controller programming is relatively sane. You can find UR bots in many schools. Take a look at AATB for examples of that avenue.

Robots are hard and fun! I highly recommend giving the field a try. It's a great time for it.