The crusty white buildup you're seeing is almost definitely mineral deposits from hard water - that's limescale. The slimy feeling is a bit counterintuitive but also typical of hard water; soap doesn't rinse as cleanly, so it leaves a film on your skin. You can get a water hardness test kit from any hardware store or online for cheap, and they're pretty straightforward to use. If you're curious what you're dealing with before spending money, that's the easiest first step. Older houses in many regions do tend to have harder water, so it wouldn't be surprising.
Before jumping to a whole-house softener system (which is pricey to install), there are some less expensive options worth considering. Point-of-use softeners for just your shower or kitchen tap run way less money upfront. You could also try replacing aerators and showerheads with new ones - sometimes that's all you need if the buildup hasn't damaged the fixtures themselves. Vinegar soaks will dissolve limescale temporarily, though it keeps coming back. If hard water is really bothering you with laundry, dishes, or skin issues, then a whole-house system might eventually make sense, but test the water first and see how bad it actually is.