I'm reading William Dalrymple's book White Mughals right now, as a followup to his book about the last Mughal emperor and the Great Rebellion of 1857, and while these are both really interesting looks at later phases of British colonialism in India, I'm curious about how the British first got established there--I gather the East India Company first held significant territory in Bengal, but it was originally meant to facilitate trade with the East Indies, not the Indian subcontinent.
So how did the British come to get a significant presence in India? What caused their focus to shift to India over the East Indies? And can anyone recommend any good books that cover general Indian history in the 17th and 18th centuries, or at least the initial period of British colonization?
This is a very big question.
England was actually a late-comer to trade in India, the Dutch and the Portuguese arriving earlier than them. As you said they were initially there for trade setting up a factory (which was actually a warehouse. The idea being they could store produce independent of ships arriving as the price of goods inevitably went up when the ships arrived) on the coast in Surat. The English did not have a significant advantage over other European powers initially and recognized the suzerainty of the Mughal emperors. There were multiple scuffles with European powers in India as they supported different India powers, playing them off one another. The British were very active in ensuring the best trade deals actively protesting tariffs on their trade and rather selfishly demanding that Indian trade, should being taxed (to provide a competitive edge for themselves). The result of the British strong arm tactics led to the Nawab of Bengal seeking retribution (with the aid of the French), by attacking Calcutta (now Kolkata) and the infamous "black hole" massacre.
The major shift in power came after the Battle of Plassey in 1757 in which the British sought retribution for the attack in Calcutta. This rather ignominious affair was a battle between a makeshift army of British and Sepoy (local men) led by Colonel Robert Clive (Clive of India) against a huge army commanded by the Bengal Nawab, Siraj ud-Daulah. However the British managed to convince (bribe) his Commander in Chief Mir Jafar to abstain from the battle (in return for becoming the new Nawab) causing a significant upset and as reward the British were favoured and created as local tax collectors.
This essentially set the precedent for the British becoming power-brokers in Bengal and having their official recommendation a weapon in vying for power.
The story continues from there as the British became more ingrained in the Indian system however this is generally seen as the turning point from the British being another trade interest to actively administering regions of India which slowly began to expand.
A few books I would recommend:
Unfinished Empire - John Darwin. This is not specifically about India however it does explain the events explained above in some chapters.
The Men Who Ruled India - Phillip Mason. Phillip Mason was an Indian Civil Servant and his books explains the background quite well however somewhat briefly.