How do you push yourself in your character design and art?

For me, when I want to draw a character in a certain style, I tend to find existing characters in that style and draw them from reference. The process of drawing each detail of the character you're looking at does a good job of getting you to see how they did certain things. Also, don't be afraid to reference and draw inspiration from those character for your own original ones, as long as you make enough different to make that character your own.

All in all, I find drawing from reference is the best way to pick up new tricks.

It requires a bit of experimentation, try different face shapes, eye types, noses, mouths, bodies, etc........don't be afraid of drawing something weird......some of your tests may not work, others may do in a less exagerated way, and others will do wonders.

The key is to have a certain variety of eye-types, nose-types, mouth-types, etc......

Mathematically speaking, your posibilities will increase exponentially for every new type of bodypart you have like this:

example: a creator that knows 5 eye types, 3 nose types, 2 mouths and 3 ear types:.....this creator would have 5x3x2x3 combinations which equals 300 posibilities for unique faces....this doesn't even consider skin color, hair style,body type or even species, so this method will multiply your options.

A good place to start is looking for artwork and artists outside of your normal art diet.

Also to look at clothing and costume references from real life and from different pieces of media (video games, movies, tv shows etc). You can narrow the field of references when it comes to genre of what ever your working on. Always pay attention to the media you watch and ask why you think the artist or costume designer chose to draw or style the character in the way they did.

Practice constructing your characters differently, like when you draw the head and body, so no two characters look too similar. There's this exercise I saw in a video by Lavendertowne where she had multiple heads of varying face shapes that she drew different facial features on. With each head , she pushed thing like eye shape, accessories, facial hair and the like so no two faces looked exactly the same.

Iterate. I personally have a hard time doing this, but a good practice when it comes to character design it to spend time on trying different outfits, body and face shapes, accessories, and defining details on your characters. Taking on different inspirations and trying them out to see what works best for a character. There's this doll youtuber that I watch, her channel is called Dollightful. She repaints and modifies dolls so that they can be one of a kind. One of the most interesting and insightful portions of her videos is in the beginning where she lays out her inspirations for each doll repaint and shows the dozens of iterations she goes through before finalizing on a doll design. It's amazing stuff and her being a great artist is a plus too.

Certainly I am not an expert to explain this BUT on my case, I am trying to get to that goal by learning step by step, one thing at a time. While making my webcomic in itself provide a lot of practice, it is not enough if I just draw the panels without aim. I have searched tutorials of how to draw bodies, searched for muscles and bones, and have added that information in every chapter that I draw, and once I learn a thing I advance for another. Sometimes is something very little as how to draw an ear, but just by that, once I have fully comprehended that information and could draw it without looking at my notes, then I know that that one panel is better made than the one before.
In resume, as by drawing your thing you will have enough excuse to practice, check some tutorials, add it to your repertorie, and seatch another tutorial, and keep repeating until desired. There is no other way that keep drawing, in the end.

I take note of what I do too often and try to break that sillhouette. Also, practicing by learning other styles and other artist's methods outside of your comfort zone can help change the way your art looks. We tend to think different tools would all look the same used by the same artist--but the tools actually do help you draw a different way. So, try out what other artists are doing. Everyone from the masters, to maybe someone you emulate who is still alive (although you can't really post those studies, they're still very helpful to do. You learn a lot by copying and trying to reverse-engineer how they did it to try and recreate their art 1-1.)

A few approaches:

One is to just sit down to sketch and like... just start with what you normally do. Then for the 2nd try, do something totally different. If you drew a tall guy with short hair, maybe do a short girl with long hair next. Then try something different for the 3rd one: maybe if both of these characters were thin, try drawing a heavier character, or one with more toned muscles. Then maybe the next one you try a different jaw shape, and the next one different nose and eyes, and so on. Experiment with shapes, go bonkers! Same can go for animal characters- if you start with one dog or fox-like character, maybe try a cat next, then a bird, then a lizard, then a fish, and so on. Or sticking within one species you can go through similar experiements: how do the different genders look? What is the difference between a buff person in this species vs. a thin person, and so on. Same with robots and whatever really.

Once you do some practice like this, then when you sit down to sketch ideas for new characters you can have a larger pool of ideas and shape language to choose from. It's easier to dedicate time to this sort of practice than trying to generate it out of thin air while simultaneously trying to design a character.

The second thought is basically the same as the above, but while making heavy use of reference. Take time just sketching different hair styles, noses, mouths, clothes, animals, etc. Try to take note of how each is different, how to structure different ones, and how to translate those elements to your style. Or different species of similar animals to see how they differ. & etc.

I have a lot of varied character designs. What usually happens for me is that I'll see an actor, or just another human being on the street, and their face will just be PERFECT for a character of mine whose design has been a little 'hazy'. I'll then figure out how to adapt their face to my style, and once I have that, the way it expresses is very much dependent on the character's personality.

Like others have pointed out, using different shapes can help you experiment with your characters' body shapes! I draw manga myself and really love when I see manga artists who are able to create interesting, unique looking characters. When I design my characters, I also try to put aspects or hints of their personality/backstory into the design. The whole iconography thing.

Whenever I see somebody in real life or in a game or manga with an interesting face, I try to think about how I'd go about integrating those features or drawing that person or character into my own artstyle. The distinctive eye shape and long face and nose of Jules and Urien in Errant was partially inspired by a school friend of mine, and partially by Belldandy from Ah, My Goddess!, whose serene eyes I thought it'd be interesting to use to create a character with a face that's hard to read.

Drawing fanart of characters who are really outside of the kind of characters you normally draw is a good exercise, and also drawing portraits of real people, because they often have face shapes or hairstyles you'd not necessarily think to draw. Doing portraits forced me early in my career to come up with rules for "how do I draw people who aren't skinny?", "How do I draw older people?".
Fanart of groups of similar characters (ie. characters who are all around the same age) can be a good exercise. For example the Sailor Senshi, Homestuck characters, the cast of Azumanga Daioh etc. because you have to think about how to depict the differences between them. So for example, here's some old art (ugh, old art...) of me drawing all the Homestuck Trolls:

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Since the characters are all around the same age, and all have black hair and black eyes and I was using a very limited black and white style, the exercise became about really thinking about things like nose, eye, jawline, eyebrows etc. to see if I could give each of them a distinctive face that captured the character.

I drew this for another thread last year, but it's a good guide to how I think about designing faces based around how I approach a self-portrait and picking out the distinctive features of a face:

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Mor me, I try to apply certain shape and use it to create my characters. For example, my main character Audra storm, is mainly rectangular with straight lines, its to represent her more serious personality when compared to her friend Cosmo S. the villain who is the only character in my story without shape language to represent his chaotic personality.

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each of my characters is a piece of my very soul

For me, having the ability to just focus on art and nothing else for short periods helps me to come back to my project with fresh inspiration. The mind needs a break sometimes.

Along with breaks, I like to set myself small, achievable goals per day. Like, my pages are hand-drawn, take a LOT of work to color, and there's no undo button (yay traditional art...), so I really have to pay attention and be careful. So I'll set myself a reasonable goal per day, like 'ink that page', or 'color two panels'. If I'm not trying to do too much in a given day, I can concentrate on doing my best with the limited goals, instead of stressing about getting an impossible amount done.

And if I finish doing my goal, and want to keep going cuz I'm in the groove? Awesome. Bonus points. I just have to be careful to listen to my brain and body when it tells me it's done for the day, and that continuing will result in... shall we say, 'declining quality'.