Great question! I actually spent my college capstone project working on an extremely interesting project focused on intelligent automation control and got to meet some brilliant engineers working in robotics. Your question has several good and correct answers but I'll stick with the one I personally think is best and if you have any follow ups feel free to reach out and I'd be happy to try and set up a conversation with an engineer in the robotics space.
- Without a significant body of automation systems experience (typically through a manufacturing job) you'll need a technical degree. The best technical degree to pursue would be in engineering and almost and engineering degree can pave the road to robotics and this is going to depend heavily on #2
- Well...what part of robotics do you want to be involved in? Engineers are problem solvers first and foremost and the type of engineer just decides where you're going to focus your problem solving abilities on the technical side of things. I'd strongly suggest taking a look at entry-level robotics positions on a job platform (LinkedIn, Ziprecruiter, etc.) and find job descriptions that make you say "that's what I want to be doing!". Education is a tool you get a hand in customizing to match your passions, so start with your passions, figure out what would get you out of bed, then take those desired role qualifications and select a major (using a college's posted class requirements for each major) that gets you that class experience. It may not be your dream class load but if robotics is your passion then to have a good chance at a career in that field you have to work backwards
I know its not a direct answer but there are A LOT of jobs that go into the filed of robotics. I've been involved with the software aspect (Computer Engineering) and robot assembly (Mechanical Engineering), but there are elective classes in systems, dynamics, circuits, controls, and materials that can fine-tune any technical degree to be suited for robotics, you just have to figure out what you want to do.