I realize I won't be building Smartsheet anytime soon, but I want to start learning the tools with enough power to get me there.
I do plan to "Start with PHP, Rails or Node and build stuff," but first I would like to have a broad understanding of the frontend to backend path you describe. I know nothing of this process. I am looking for the basic concepts. In some ways, I feel like I'm trying to take derivatives with a fuzzy understanding of algebra. Following the book's example can get you from x^n to n*x^(n-1), but not to understanding.
It's like reading documentation for MS Word, with every menu item explained but no clue as to how you'd put together a large, complex document. How to use every little feature, but nothing on when and why to use the features.
Please keep that in mind, writers. :)
https://wappalyzer.com/download - one of my favorite tools - will give you a little more insight into what smartsheet uses - they're homepage at least, is a LAMP stack with Drupal.
Smartsheet would be a tall order for your first or 5th project. Get a cheap server somewhere - start with PHP, Rails or Node and build stuff. Learn how to communicate from the frontend to the backend - make mistakes, rinse repeat.
I haven't bought or read a book in years; once you get the basic concepts down, searching google / stackoverflow for how to do something will come much easier.