I've been reading about how the British empire came about, and along the way, I've noticed sources just seem to switch from talking about the East India Trading to talking about the British in general. Was it the British empire, or the East India Trading Company's empire? Surely a merchant company didn't just decide to conquer the world.
There was no such thing as the East India Trading Company. I assume that you are referring to the Honourable East India Company? Or, earlier, the "Company of Merchants of London trading with the East Indies". The HEIC was only active in the Indies, but it was under a Charter from the British government which gave it full authority over the territories that it controlled, the right to raise armies etc. However, by the mid-Eighteenth Century it was becoming, effectively, a branch of British administration, acting as an agent, as it were, of the British government. Acts of Parliament in 1773 and 1784 confirmed this. Certainly by the early 19th Century it was hardly a commercial organisation at all, but was really Britain's proxy administration of Britain's Empire in India. After the Mutiny the Government, in effect, nationalised the HEIC, taking over control of the HEIC's Indian interests, assets and powers. Finally in 1874 the HEIC was formally dissolved.