There are two things that you are mentioning:
1. Hard water: Lime buildup on fixtures
2. Chlorine residue
Both can also cause dry skin and hair as well.
Hardness and chlorine are treated using two different methods:
A water softener is the ONLY thing that will remove hard water minerals (the lime on your fixtures).
A carbon filter will remove chlorine, taste and odor, and many chemicals you may not want to drink but does not remove hardness.
A reverse osmosis filter can also provide very pure drinking water - but realistically it's limed in the amount of water it can filter (a few gallons a day). You aren't going to install a whole-house RO filter, for example.
A good water softener can be purchased online for $800-900 with a Fleck control valve. It has to be sized properly to your hardness levels and also to the amount of water you use. A local water treatment company can install one for ~1,500 or so. I would start with a water softener, and then see how your skin and hair feel.
Avoid Kinnetico, they are ridiculously expensive and provide no real value beyond what a standard softener provides. Also avoid anything that calls itself a water "conditioner". A water softener is a specific thing. A water conditioner could be anything. Avoid anything that involves magnets, electro magnets, or is "salt free".
If, after installing your water softener, you still want a carbon filter for drinking water, best to install one under the sink on the cold water line.
Keep in mind that both the softener and potentially a carbon filter will require maintenance. The water softener will require salt, and the carbon filter has to be replaced about every 12 months.
Q