First and foremost, you just have to keep drawing. Even if you don’t want to draw realism, draw from life. Do at least a drawing a day. Do that for a year. Do that for a few years. Keep doing that. Think about stopping. Then keep drawing again. I’m still on the first step. I don’t know of anyone who has finished the process. Well, I mean, unless “they died” counts. If you want to draw cartoons, you still need an understanding of basic anatomy and proportion and stuff — outside of a “how to draw book” — to make a solid character or a believable environment, fantastical or grounded in reality.
Secondly, and I’ve been skimming through this book, check out The Animator’s Survival Kit. Even if you’re not animating, it’s written by a man who worked on Who Framed Roger Rabbit?. It’s really a solid book for teaching how people look when they’re in motion, when they’re at rest, when they’re winding up to do something, the works. It’s just a great book. Look at Disney drawn animations and cartoons as well for good cartoons. I really think this book helped me loosen up a bit, even if I’m not doing cartoons all the time. If you’re looking for books, I’m going to say avoid looking for anymore “How to Draw X” books, like “How to Draw Dragons” or whatever. I say this as someone who started with a dragon book, and drew them all. And then uploaded them to DeviantART. And then realized years later that’s why I felt so stiff when drawing and that I wasn’t improving. Because I felt like I needed a guide. If you’ve been using them a lot, it’s even more important to return to the first paragraph, which is, in summary: Draw. A lot.
References are fine to use. Hell, I use them all the time. But don’t copy them directly via tracing them or color-picking from them or whatever. You have to do that all on your own. It’s going to look wonky and bad at first. Everyone has a million bad drawings that they have to get out of the way before they can do one good one. And then they do a million more bad ones. Tracing is great for understanding, say, how Joe Maduriera did his linework for Darksiders (if I can throw out a personal favorite), but it doesn’t give you any boost in your skill. If you eyeball it, try to imitate by reference but make it your own thing, you’ll learn so much more, not only in how to work at that specific moment, but you’ll have a memory for the future for how to do that again.
For drawing animals, the same thing applies. While a small mistake isn’t going to be as noticeable on a dog as it is on a human, you still want to draw them from life. If it’s a cartoon animal you’re drawing, you’re going to really want to try and understand how the skull and head is formed, I would think. From there, you can take the body and exaggerate how you need to. If it’s anthropomorphic like Mickey, well, then you’re back to human anatomy and stuff like that. So just keep on drawing from life. Once you have that collection of life drawing in your head, drawing imaginative images becomes easier over time.
TL;DR: Draw. Draw? Draw! Draw. Keep drawing. I didn’t tell you to stop. Draw some more. Little more. Little more. Mmkay, good. Now, draw. Draw. Draw some more. References are cool. But draw. Draw. Draw. Animator’s Kit is a good book. Has some techniques for animation and cartooning. But you have to draw from life first and foremost.
These are my own opinions. Take them with a grain of salt.