I think there are two very broad main areas of research in this context:
The first is what could be called "hardware" - how people lived their lives, the social structure of the time, the law, the jobs they did, how they dressed, what they ate, the homes in which they lived, how they passed their leisure time etc. To use your own example, whether there were prams and, if so, whether your character could have afforded one. One should know enough and research enough to avoid anachronisms, even in small details. A great deal of such research, while time-consuming, is fairly readily available.
Perhaps too often, writers limit themselves to researching the hardware, but the second area is more difficult and, to my mind, equally important, if not more so. It concerns how people thought, spoke, behaved and interacted - the social mores and conventions of the period in question - call it social software. Since most stories centre on people, this second area is, I would suggest, the more important of the two to get right. Perhaps the best way to get a feel for and knowledge of such things is to read contemporaneous works of fiction. So, if your setting is early-Victorian England, read early Victorian novels set in their time and, if you can, try not to read for the plot and characterisation, but for what the books tell you of the way people thought, spoke and behaved at that time. The same applies to any period in which contemporaneous fiction was produced. Unfortunately, prose works in English really only get us as far back as the mid-1700s. Before then, one can glean a great deal from such sources as newspapers, pamphlets, diaries, letters and plays. I stress contemporaneous as historical novels are written by people who have faced the same problem that you now face; and they may have cut-corners, or simply, made mistakes.
A typical example I came across recently was in a historical novel set in the nineteenth century, in which, following a skirmish, an English officer, when asked the whereabouts of a man he was with, replies "He didn't make it."
Bad enough to have a young lady of good family in 1890 walk down the street in a hobble-skirt (a fashion fad only popular for a very short time between 1910 and 1913) - hardware. Perhaps even worse to have her engage in conversation with a young man to whom she has never been introduced - social software.
As for "How much research do I need?" My answer is: enough not anly to avoid anachronisms, but also to bring the period to life and have your characters act, behave and speak in a manner true to that period. Unfortunately, very often, this means a huge amount.
