FPGA is pretty challenging to get started with. If you decide to go in on it, you're going to spend quite a bit of your time at the beginning getting your ass kicked. Vendor tooling blows, open source tooling is playing continuous catch-up or is much more focused on VLSI, and resources for beginners feel pretty anemic compared to what you're probably used to coming from the software world.
I can't tell you if this is "the way", but I can tell you how I started. I flipped a coin to decide between Altera and Xilinx and started with a Terasic DE2 board (an Altera Cyclone II devboard) that I borrowed from my university. I don't recommend using this board or something like it (it has actually been superseded by a nearly identical board with a Cyclone IV in place of the dated Cyclone II); the extra peripherals are a headache more than anything, and the simple breakout pins on the Terasic DE0-Nano are greatly appreciated. As for the environment, you can go and download Quartus from Altera basically no-questions-asked.
After pissing with the board for a bit, I decided to pick up a book. Rapid Prototyping of Digital Systems by Hamblen, Hall, and Furman is good, if not a bit out of date. I had this book to thumb through instead of read front-to-back. It is written for older versions of Altera's Quartus software, but with a little bit of exploring the UI I was able to find just about everything I needed. It makes a decent quick reference for Verilog and VHDL syntax, has quite a bit of info on interfacing various things to an FPGA (think PS/2 mouse, VGA, etc.), and a couple chapters on microcontrollers and microprocessors.
An important bit to know is that that a lot of the usage of an FPGA (at least in the way I use them) happens in two phases; the actual logical design, which you'll do on your computer via simulation, and the actual usage of that design on the FPGA. The logical design happens largely in ModelSim (with Quartus only being used to generate libraries at this stage), or some other Verilog/VHDL simulation tool. Altera ships ModelSim free-of-charge with Quartus. It is your run-of-the-mill Tk-based clunker of an EDA tool, but it works and I haven't had it crash on me yet, so I can't really complain, even if I have some gripes with it. This is where most of the work/magic happens, at least for a beginner; you write your logic in your editor of choice, write testbenches, and simulate those testbenches in ModelSim. Having read the Harris and Harris book mentioned in my initial post in the past, I took a quick look at a processor called "Ahmes" on GitHub, and just had a go at it. After getting a primitive processor working/executing instructions, adding some peripherals like some timers is where things started to come full circle, and I started realizing both the power of the FPGA, and why things are the way they are on a microcontroller/processor.
The bit where you actually put it on the FPGA hardware is largely uneventful, or at least it was for me. I didn't have multiple clock domains or anything like that, so I didn't suffer from any timing issues, and basically had the good fortune of just mapping pins and letting it rip. In theory, actually translating the design to the chip doesn't do much, but in practice, a design that exists only on your computer isn't terribly useful. Actually using the hardware adds that "visceral feeling" and lets you play with what you've spent so much time doing, along with getting you familiar with the process if/when the day comes that you actually need to do it. You also get to enjoy/experience the power of bolting on arbitrary hardware to your jalopy processor for the sake of VGA output, GPIO, timers, SPI communication, etc.
I wouldn't consider myself terribly talented or knowledgeable, so if you just throw enough time at it, you can probably end up having just as much fun as I did.