Keep in mind that if all goes well, and you gain entrance to med school on the first shot, you'll finish your psychiatry residency when you're 44 years old. Just something to think about.
Also keep in mind that the majority of students entering medical school will change their minds as to their specialty after their clinical rotations. What you may like now might not be what you'll like a few years from now.
You didn't mention why you want out of the law field. That can be good fodder for a personal statement.
To get into a med school, you'll certainly need some medical-type exposure. Preferably something not primarily in a lab. As you mentined, EMT is a good way to start. Others more psychiatry related can be volunterring in a psych ER, or by shadowing a psychiatrist. Also, you can try volunteering at a psychiatric hospital, though these positions tend to be difficult to procure due to privacy violations.
Do lots of reading, read some journals, and certainly try to observe a psychiatrist at work. It'll help you get a better feel for what you'd like to do.