Pull faces.
No, really. Look at this dude;
This is Ward Kimball, a super good animator who really influenced my own work, especially in the expression department, but almost all animators do this. You want a serious piece of advice? Grab a mirror, think about your character’s motivation and just get silly.
I’ve always had quite a rubbery face and people constantly tell me that I am a ‘cartoon’ of a person, but you don’t have to be extrovertal in real life to be able to pull faces back at your desk. Sometimes you don’t even need to be looking at yourself for it to help with drawing expressions; just feeling how the muscles of your face move helps, and it gets you in the mind set of the character. A lot of people do it naturally- ever found yourself drawing something and pulling the same face as the person you’re drawing? Good. Keep doing that.
Animation is acting, you just do it through drawings; illustration or even just scribbling a character with a certain emotion is just as involving and important for me. If you want to draw a character feeling sad, feel sad. Don’t just imagine them being sad, think about what would make them sad; understand why it would make them sad, and in what way. There’s a thousand ways to be sad and a thousand ways to look each type of sad. Don’t be simple, don’t be complicated; be deep. Know what you are drawing from the inside out.
The very word 'character’ tells you that the subject you are drawing, be it a human or a robot or a shoe with a face, has a personality. There are all sorts of tutorials out there about the anatomy of the face and typical expressions, which are really useful and I do recommend looking up, but for me the most critical part of the process is what is inside; realizing that you are drawing something with emotions, not just trying to draw 'an’ emotion on that character’s face. You have to understand what you are drawing. What is an emotion? A reaction. What are they reacting to? Why are they reacting that way? Make it mundane or make it dramatic, but give your character a reason to look the way they do.
Characters are people. Respect them for it, and they’ll help you find the expression that you’re looking for.