What does Mr. Shimerda want Jim to do for Antonia?
Mr Shimerda wants Jim to teach Antonia to speak English.
The passage is in Book I, Chapter 3:
We went with Mr. Shimerda back to the dugout, where grandmother was waiting for me. Before I got into the wagon, he took a book out of his pocket, opened it, and showed me a page with two alphabets, one English and the other Bohemian. He placed this book in my grandmother’s hands, looked at her en- treatingly, and said with an earnestness which I shall never forget, “Te-e-ach, te-e-ach my Án-tonia!”
Mr Shimerda, who spoke only Bohemian, depended on Peter Krajiek as his interpreter. None of the Shimerdas could speak English -- "They could not speak enough English to ask for advice, or even to make their most pressing wants known" -- so they were completely at Krajiek's mercy. Mr Shimerda's plea for Jim to teach Antonia is both a token of his trust in the Burdens and Jim and Antonia's budding friendship, and a ploy to become independent of Krajiek.