Just moved into a house with horrible water. How to fix it?

Sounds like "hard water". You can get a faucet mount filter for around $30, with a built in switch, so you use the filter only when you want, making the filter element last longer, or you can get an undersink filter which is trickier to connect, but is a better filter. Or you could just buy a brita pitcher filter, and use it for filtering your final rinse water for shower/shampoo.i