You could well live your whole life without needing anything more than Newtonian Physics. For most of us, relativity is a fun thought experiment. If you want to grapple with it, special relativity is the answer to "how can the speed of light be constant regardless of the speed of whoever is measuring it?" In his "vulgarisation" books, Einstein explains it with nothing more sophisticated than trains and stopwatches.

General relativity is more complex and quickly goes in complicated mathematical weeds but is just as profound from a philosophical point of view, which is that things do not merely affect other things around them, but instead change space-time itself. You can see with a couple of clicks observations of phenomena predicted by it, like black holes and gravitational lenses. It’s interesting to think about even if you are not directly affected.

Once upon a time most households had a small particle accelerator, used daily. While the the electrons in the cathode ray tube (CRT) traveled at relativistic speeds (something like 0.1-0.3 c, from what I can tell), people did not need need to know about special relativity to change the channel on their TV.

That said, those effects would have been small, and likely handled in practice as "some arbitrary consistent (or random) error."