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.