What’s the best route to go to get into CID?

CID is not usually an entry level MOS. You have to apply and be accepted. Assuming you meet the standards, just apply and get your acceptance letter at your first duty station. You still have to wait around until you get a top secret clearance. The whole thing should take about a year. As long as you are accepted, it's a very easy MOS to reclass into.

I suspect that you would need some Criminal Justice education.
Or get some MP experience first reaching E4 - E6, and at the same time, work on some self help military courses. Just a thought.

CID is not entry level, and you must be at least an E-4 to submit the application. You should contact your local CID office and see what the requirements are, they can help you and may even get you an intern position to see if it's something your cut out for. Autopsies usually get to people. I was an MP, and they have an easier time moving into CID because of their previous training. For instance, I didn't have to do the internship because I was MPI for two years and worked closely with them on many cases. I had extra time to meet the 60 hours of college credit as long as I promised to complete the remainder needed in a certain amount of time, which I did.