A tip about drawing eyes, in a stylized way.
First thing that makes a lot of difference is making the top eyelid thicker. This represents the eyelashes and gives the eyes more volume. It's also important to remember that the eyebrows react to where the character is looking, like the eyemask I have mentioned before. So adding this will give your character's expressions a lot more life. Exaggerate expressions! If you want a more cartoony
look, you must exaggerate to a point that it's a caricature. Push it. And always remember the basic shape of the realistic eye, and try to stylize it based on that anatomy. Stylizing things when you understand their anatomy will turn out a lot more believable.