There are very few things that China should learn from India. To be frank,most of the time, it's the other way around. Anyway, here goes ....
1) The right of the citizen. Despite rampant corruption,mafia, gangs and political influence, one thing that hasn't changed for the past 65 odd years is the right of the citizen. The citizen of India is supreme. The constitution protects him to the hilt. He can boldly take the government to court over what he feels is an injustice to his life or to the society. The Right to information act only makes him stronger in today's world.
2)To respect one's culture or religion. Indian is diverse. Very diverse. However, we unanimously consider ourselves as Indians rather than any ethnic or religious community and for the most part are united when it comes to international issues. There is also a growing sense of respect to the fellow human being which bodes well for the entire country.I think this is also present to a certain extent in China.
3) To have a logical dialogue over every single issue under the sky. We Indians do that a lot. In fact, it has been ingrained in us from the ancient ages. We are not afraid to voice our opinion on a huge range of topics. You can find people discussing and arguing for their points of view at all strata of the Society ( from television stations to the local tea shops)
Regarding English proficiency, it's nothing great to speak of. A majority of the Indians, though they speak English, are rather poor at it's usage. Nevertheless , 300 million + people speak it with an acceptable level of fluency. You will be able to notice vast differences in the fluency and accents, as you move from the North to the South. A majority of the schools in all Big cities and Towns have English as their principal mode of instruction. Colonialism and servility for over 200 years have left behind a rather ugly scar on Indian society. The scar i'm speaking abuot, is that of people considering Fluency in English as a sign of good upbringing and social progression. This in turn provides an impetus for Indians to learn the language.
Everything else, I feel, India has to learn from China. Infrastructure, Education, Politics and Governance are some of the issues that I can think of.