I also come from a non-technical background.
I spent my early career in Management Consulting and Product Usability Testing. These two areas set me up well for a career in product management.
That said I’ve found a few things helpful.
Learn SQL – This was my first foray into the technical side of products. Its a great place to start, because its immediately useful for product managers. You can start to understand the data model and be able to pull your reports.
Take Treehouse courses (Start Learning to Code with Treehouse) – I took some classes in Objective C and Python from Treehouse and found them really useful. I never finished the courses, but they gave me good exposure to a few of the technologies we used at work. You can also try Coursera, Edx, Udacity and Udemy, but I’ve not tried them personally.
Read The Mythical Man-Month – Its a classic book on the realities of software development and a great read for non-technical PMs.
Learn from your dev team – Most engineers are happy to explain what they are working on. Interestingly, it also happens to be a great way to build rapport with the team. Over time you’ll be amazed about how many important concepts you can absorb.
As a PM its more important to understand the concepts than to master a particular language. Focus your energy on technical awareness/exposure, not mastery.
Disclosure: The Treehouse link is a referral link. Feel free to not use if you prefer. I only refer people to products I actually use and love.