If you grew up outside the Telugu world, your first encounter with Telugu storytelling was probably through Tenali Rama. That's a good start. But the Telugu tradition is older and wider than the trickster tales, and some of its best stories are the ones that haven't yet been collected on the same shelf as the more famous ones.
The trickster: Tenali Rama
Start where most families start, because it works. Tenali Ramakrishna was the poet-jester in Krishnadevaraya's court, and his stories are short, self-contained, and funny in a way that survives translation. A cat that learns to fear warm milk. A thief tricked into watering the garden all night. The small clever move beats the big forceful one, and a child clocks that before you've finished.
For a non-Telugu child these land the same way they land for a Telugu one, because the joke lives in the shape of the story, not the language. Start here, at three or four, and let the child ask for the next one.
The boy who held the line: Prahlada
Bammera Pothana's Telugu Bhagavatam gave Telugu children one of their most-loved figures: Prahlada, the boy who wouldn't stop saying the name he believed in, no matter who told him to. It's a story about a small person holding a line against a very large adult, which is something every four-year-old understands from the inside.
Share the boy and his stubborn calm. The fiercer imagery around him can wait a few years. The part a young child needs is the part where the smallest one in the room turns out to be the steadiest.
The verses children memorise: Sumati and Vemana
Telugu has a tradition non-Telugu families almost never meet: short moral verses, the Sumati Satakam and Vemana's padyalu, that Telugu children learn by heart the way English children learn nursery rhymes. Each is two or four lines with a small, sharp point about kindness, patience, or not trusting a flatterer.
You don't need the Telugu to use them. Say the idea in the language your child speaks, then tell them children in Andhra have recited this exact line for five hundred years. That fact alone does something a lesson can't.
How to pick, and how not to over-explain
Pick for the moment, not for the heritage. A Tenali Rama story your child asks to hear again is doing more for them than a solemn classic they sat through once. The Telugu-ness is the setting. The story is the story.
And don't translate the culture out of it. Leave the name Krishnadevaraya long. Leave the pot of curd a pot of curd. A child will ask about what they don't recognise, and the asking is where the tradition actually gets passed on.