There are free academic studies to help you learn about the *diffusion of science into common use, both for scientific communities and the public. See, for example, The Diffusion of Scientific Innovations: A Role Typology, or this article The science of diffusion and the spread of public policy
If you use a search engine; you may have to go to later pages because the scientific concept of "diffusion" e.g. a material in a liquid will dominate the early hits.
Here is a Wiki on Diffusion of Innovations that may be helpful, and another on The Rising Speed of Technological Adoption.
50 years from inkling to saturation.
As a general rule; following the laws of adoption given in Diffusion of Innovations (which do not specify a timespan but how different types of people vary in the time it takes them to adopt new things), but figure it takes 50 years to get from "cool scientific discovery" to full exploitation in modern society (whatever that means for the particular discovery).
Beyond that fifty year span; the tech is up to you. The tech of 200 years from now (if society does not collapse) will appear to be like magic. But in science fiction you must root the explanations into something scientifically plausible anyway. For example, a compact fusion reactor the size of a pocket watch is actually plausible if we have some technology to control atoms and force cold fusion of individual nuclei; the force required to do so is minuscule if it could be focused. Hot fusion relies on random chance and the statistics of large numbers; heat atoms up and they vibrate wildly, contain them somehow and their vibrations will cause some of them to collide with enough force to fuse and release energetic particles. There is no explanation in Star Trek for how a replicator works, but in particular it would never be attributed to "magic," the writers would make something up about using a template to fire certain atomic ingredients into the right places to form chemicals and compounds that are present in an ice cream sundae or a rib-eye steak.
So you just need to develop some sense of what is scientifically plausible BS and what is not!
For recent scientific advances in all fields; I'd recommend New Scientist; London based UK monthly magazine. Costs money. I've subscribed to a dozen such magazines and that is my favorite by far for clear explanations that don't trigger my bullshit detector.