I think it depends on what your species is a bit, but there are plenty of ways to express your furry side in public!
The first thing you can do is collect/wear things relating to your species. That's the easiest thing to do if you worry about anyone judging you (in my opinion you shouldn't, but some people are more shy about these things
).
I'll bet you can find someone to customize clothing for you too. An easy way to start is by designing a costume sweater (cuz you can wear those out casually too, if you want to).
As for behaviors and gestures, what I do is think of/research the gestures and behaviors of my species, and see how I can incorporate them into my everyday behavior without getting weird looks.
Being a rat, I tend to quietly hang out in the background most of the time. I imagine a liontaur would have good posture and walk with pride. Traits of both big cats and hoofed creatures would apply. Every creature has their own body language.... research it and see what works for you!
You can also subtly (or not so subtly, depending on what you prefer
) throw words that relate to your species into your everyday conversations. For example, you could call your family or your group of friends your pride, since you're part lion.
That's kind of all I've got for now, but I hope I was able to help you start coming up with ideas!
