I got into the embedded world a few years before Arduino and others like it. I'm sure there were easier approaches, but the way I learned was by putting a PIC16 into a breadboard and going from there. I have always since followed this approach:
1) Determine what parts of your project will need microcontroller support, such as displays, audio codecs, memory, etc.
2) Determine exactly how many GPIOs you will need, as well as specific microcontroller peripherals.
3) Go to DigiKey or any distributor whose website has a parametric search option and start browsing for controllers that support those requirements. If, as an example you needed to support all three devices mentioned in #1, you might want a chip containing an LCD controller, I2S, and memory interface that fits your memory type.
4) You'll be left with (hopefully) a choice of a few chips from a few different vendors to choose from.
5) It now comes down to package size and cost. If it doesn't impact the space and cost too much, I like to choose a chip that will leave me with a handful of unused pins. Your project, when all is said and done, will not work EXACTLY the way you expected when you first designed it, and you will thank yourself for the extra GPIOs down the road.
6) Now that you have chosen a controller, it's time to create a schematic of your entire project. This will ferret out problems like discovering that your controller has two needed peripherals multiplexed to the same pin(s), discovering that you don't have enough GPIOs, etc. Make sure you thoroughly read the datasheet sections relevant to the uC peripherals you will be using, as well as the datasheets for any external devices. If you have done that, your now finished schematic should be accurate enough to move on to prototyping.
7) If you will be prototyping by hand, try and find a package you can work with. DIPs are becoming more and more rare these days so learning to do SMT soldering could help a lot in this regard. There are many ways to approach building a new project, which I will not go into here.
If you've never used the controller you chose, start by getting an LED to blink. If you reach this point then you can continue on to designing in the other modules into your project. If you're breadboarding, don't connect everything together at once, integrate each item one by one.
9) On the code side of things, it is just as important to constantly refer back to the datasheets in order to get your peripherals working properly.
NOTE) If you're learning something new, try to avoid libraries that do all the hardware work for you. After all, isn't that why you wanted to move away from Arduino in the first place?