I transitioned into being a freelance digital artist full time about 7 years ago. I've done a lot of trial and error, searching for programs, testing and choosing solutions for everything from digital pen tablets to use, to programs to help me animate. I'll include my tips below so that if you're looking to start and excel at digital art, you will have the huge head start I wish I had. I'll include some pictures and please reach out if you have questions to help you answer that question of how to become a digital artist.
Adobe Creative Suite
The first and biggest tool is the Adobe creative suite. It's a hefty $53/month but, besides my computer and tablet, it is continually the most valuable tool for my processes. They're updating their programs often and they continually become more powerful.
Photoshop I use Adobe Photoshop the most, drawing with my Wacom Cintiq 22HD directly into Photoshop. It is a pixel based program and works very much like drawing on a paper or painting on a canvas, but with the power of layers, undos, and a vast array of continually evolving brush options. You can choose to draw with what appears like a 6h pencil, or paint with what looks like an oil paint brush. If you only need Photoshop in the Adobe Creative Suite you can get it for $20.99/month. Like the rest of the Adobe programs they are constantly updating features and as long as I spend 5 minutes looking at their info on the updates, I'm continually added new capabilities to superpower my workflow.
I use Adobe Illustrator for vector-based graphic design, like logos and word art. Vector-based means what you draw in Illustrator are based on equations that make up your lines and shapes. So, unlike pixel-based programs like Photoshop, in Illustrator you can expand your drawing to any size on not get pixelated. The lines recalculate to be fine-tuned and exact at any size. You might ask why would you not always use illustrator if it never loses resolution. Well, unfortunately it requires you to close shapes and it gets complicated which shape is on top of the other. Sometimes when you delete one point in a shape, suddenly your entire screen is an abstract collage of triangles. It's also difficult to do advanced textures and textured lines with it, like you can with brushes in photoshop. It does make it easier to update all lines to a certain weight at once, and to make perfect shapes fast. So you have to choose the right program for the job.
The next most often used program depends on what I'm doing most at the time. If I'm doing book publishing I use Adobe Indesign. It is specialized for doing multiple page layouts. You can set up templates that you can apply to your different pages and it takes care of the page numbers automatically. It is really good at exporting with bleed options and gutters and margins.
If I'm doing a lot of animation Adobe After Effects is my tool of choice. You can also do some basic animation in photoshop (turn on the "motion" workspace) but After Effects is much more specialized and powerful for the task. You can lay down tracks of sound and manipulate your visuals over time to create music videos, animations, motion graphics, and tons more. It's a difficult program to learn because they have just packed so many features into it, but once you have it, it's crazy powerful. It is the most computer-memory intensive program I use though, so make sure your computer is fast if you're animating a bunch of parts. You can make your 'assets' in Photoshop or Illustrator and import them into After Effects to make them move. This is an advantage to having your whole workflow within one family of products.
Adobe Premiere Pro is for editing video. Like with the other programs you can force them to do things they weren't really built to do, but you quickly see it is more optimized for film and video. It renders faster and has much quicker keystrokes to clip and stretch imported video.
I just got familiar with Adobe Audition (they often add new programs to Adobe Creative Suite too!). It's great for recording audio. Super powerful.
I've heard Gimp is a free alternative for Photoshop. I think it would probably be enough if you aren't doing art professionally. Even if you are, it might be enough. I don't know enough about it.
Digital pen pads
You might ask, why don't I just move my finger on your computer's track pad. The advantage you get with the digital pen pads is that you can use pressure sensitivity (pressure sensitivity allows you to make a line that is thicker and/or darker). Plus holding a pen is a much easier way to get the curve you want. It's pretty hard dragging a finger across a trackpad that often has different levels of oily-ness/stickiness, and wasn't the way you grew up drawing.
I can also draw on paper and scan it in but if I'm going to be making thousands of drawings, that's a lot of scanner issues I don't want to deal with if I don't have to. I'd rather create it directly in my program, and get over that initial discomfort of drawing on a screen in order to do it. It's also difficult to remove the paper texture if you scan it in, or take a pic. Plus by creating it directly in your computer you can tell the canvas to be any size you want, and use any color, and use any pen cursor shape. You don't have to have shoeboxes full of tools anymore!
Wacom is the foremost provider of products on which you can draw as an input to Photoshop, or other drawing programs. I've heard there are some new competitors but not enough to make me reconsider buying Wacom. There are two main types of pen pads. Both come with a special pen that has a plastic tip and a pad/screen of some sort.
The Wacom Intuos product line is a pad with a blank gray surface and as you put your pen on it your cursor moves accordingly in your drawing program on your computer screen. The difficulty with this as a digital artist is that we are used to looking at our hand as we draw and it can be really difficult to get the right curves and shapes drawn in your program when you can't see where your hand is going. So difficult that I don't even try to do it. I encourage any serious artist away from these in exchange for the Wacom with a lighted screen..
The other line of products is essential for me. It is the Wacom Cintiq line. This is the type of pad that you draw on which also has a screen on it. This way you can watch your hand move right above where the mark gets laid down.
The new iPad Pros are great. It's like a Wacom Cintiq. And now, Adobe Photoshop is available in a limited version so you can start a photoshop drawing on your computer and save it to the cloud so that you can open it on your iPad. I hope they keep adding features to it because I've hit some sad limitations within it still. But it's enough to make some dang good art.
I hope this gives you a leg up! If you have any questions, feel free to reach out. And if you like my stuff subscribe!