“Imagine you are a layperson with a basic liberal arts education who wants to learn quantum mechanics on his/her own: What steps should s/he take?”

From Quora,

The first book in college (as a physics major) I used for Quantum Mechanics was the book by Griffiths.

Introduction to Quantum Mechanics (2nd Edition)

David J. Griffiths: 9780131118928: Amazon.com: Books

Seems like you can get a paperback copy for under $20, hardcover is a bit more.

You say you have a college-level overview of math/sciences. I’m not sure exactly what that entails. Does it include calculus?

I took the class with Griffiths book in my third year as an undergraduate.

A book that will help you understand the necessary mathematics is,

Mathematical Methods in the Physical Sciences

Mary L. Boas: 9780471198260: Amazon.com: Books

This is assuming you know some calculus. If you do not know calculus, you should also get a calculus book. There are quite a few options. Also some lessons about calculus on Khan Academy.

You should also know some chemistry. I had only taken a single quarter of chemistry (10 weeks) in college before I took Quantum Mechanics. Later I took a second quarter.

I read through the Griffiths Quantum Mechanics book the summer before taking the class. It’s interesting and you can understand a little by just reading it. But you need to work the problems to understand it better. Having that book could be a first step since then you can know more of what is involved in learning the subject.