First, if at all possible, never wash in; direct sunlight, if the sun/temp. causes the paint surface to feel "hot" to the touch, and never immediately after driving.
Second, your water could be "hard water" which would cause spotting.
You might want to look into a water softener.
To remove the spots, assuming they're not too bad, first try going over the spots with some wax. If that doesn't work, move onto clay. If that still doesnt remove the spots, you may need to try a paint cleaner with a mild abhrasive.