I, too, have occasionally struggled with writing even something as different as anthro characters. Much as I loved Looney Tunes and TMNT as a kid, there is a definite disconnect in my mind now that I am older. I think kids can better get into the idea of non-human creature characters just due to the visual appeal, the novelty, and possibly some of their own lack of human experience in the world, hence why the vast majority of distinctly non-human characters in fiction are found in cartoons and video games, while most adult-oriented fiction still uses aliens and fantasy races that are basically 90-99% human.
Even that said, very rarely will you find a non-human sapient creature who isn't still basically a human in their mind. Christopher Johnson (District 9) is an upright walking bug creature. He still has a human thought process, with motives, desire, and empathy that resonate with a human perspective. The talking horses of the My Little Pony universes don't even have hands and live in a mythical universe where the very laws of nature operate differently than ours, yet utilize distinctly human architecture, have human past times, and personalities leading to a very human society. And I read several different books as a kid that featured other talking animals (the Bunnicula and Mouse with a Motorcycle series, for example), where the main characters are talking cats, mice, and dogs, with human trappings.
So, I think you can write non-human characters, and still make it work and be relatable as long as you stick with a human-relatable perspective. I mean, superhumans in a lot of cases aren't really human, either, but plenty of people can understand the motivations and experiences of Superman, even if the actual details are off the human scale of experience.
As for how to introduce them, I would say go ahead and make the creatures whatever you want, just go easy on the jargon. If you introduce a new type of creature, say a warovil, be sure to give details of the creature when it's introduced in familiar terms. Try to not use inexplicable local slang too much until well into the story, or mix the new terms with only ones in such a way that the context is easier to figure out:
The two looked over the zik, deliberating on whether or not the animal was worth the purchase. Kashni looked to Zeka, noting his fuaro companion's skeptical expression. Zeka looked back to him and muttered, "That is the ugliest zik I have ever seen."
"Yeah, well, it's all we can afford," said Kashni, his claws tapping the coin purse on his belt.
"Seriously, this thing is so ugly, we could break every mirror in the city just taking it for a walk."
"Yeah, I get it."
"It should come with a free paper bag helmet. Not for it, because it won't cover it enough. For us, because no woman is going to talk to us carrying that thing arou-"
"Rao's Balls, I get it already!"
Alright, clearly you want to write a better example than that, but you want to kind of slowly sprinkle the weird details in.