In general a RO system separates the water in 2 outputs. It takes tap water and most of the minerals go out in one output that is typically dumped down the drain. The other output is soft water with most of the minerals removed.
The simplest system connects to a facet and and waste water hose placed in the drain of a sink or shower. The software output goes into a bucket or plastic trash can. These systems can produce 50 to 100 gallons per day.
You can get such systems from Amazon.com for about $60 to $80. This is the easiest option for you.
Smaller systems (about 20 gallons per day) can be installed under a sink. Due to the slow production rate these system typically store 1 to 3 gallons tsoft water with a smalll dispenser for drinking water or for cooking . Larger systems can be installed for substantially more money. Most of the time RO system are not sized to filter all the water used in a home.
Note RO systems come in 3 or 4stage versions. The 3 stage is the basic RO system. 4 stage systems add an addtionalcartridge with a resin.This aditional filter is similar to the pillows you have. This adds an extra filtering get thee water as clean as possible.. You don't need the4th stage and the extra cost to resolve your issues. Also some of the wast water can be mixed with the soft RO water to reach a specific GH if needed.