I am a very amateur student of history, and I feel somewhat ashamed of how little I know about the British Empire in India. What would be one or two good histories, accessible to the layman, that might give me a better sense of my home nation's history on the sub-continent?
For the earliest years (17th Century), a lot of source material is publicly available online.
The earliest forays into the East Indies are found in the relevant Calendar of State Papers, Colonial Volumes found here. Or you can look at images of the individual volumes 1513-1616, 1616-1621, 1622-1624, 1630-1634.
Next, there are the English Factories in India, which documents from 1618-1669. Most of them are either here or here.