What are some home devices to monitor water quality?

Keep water in transparent vessel for24-48 hours and check the odour, color, turbidity etc and confirm if any oil like substance is floating on water or not.
If these all passed, then that water may have quality but it does not ensure100% quality of water.
You can go for chemical and biological tests for ensuring100% quality water.

Many devices like(multiparameters seeba) or spectrophotometer   for any cation and anions

and other for physical test like PH-Mtere, TDS-Meter,DO-Meter

regular check of the odour,taste and purity of water as a start.
over time you need to check your electrical appliances such as kettle,washing machine and your plumbing pipes for any scaling,corrosion or change in colour.
alkalinity of water make it cloudy ,contribute in scaling and make it hard to lather the soap.
when the water is acidic,it cause corrosin to plumbing fixtures.when it is heated it cause stains on plumbing pipes,red stainsif galvanised and green if copper.
presence of chlorine is essential to disinfect the water and kill the germs that make us sick,excessive amount of chlorine can affect the taste ,odour of the water and can irritate your eyes and skin.
presence of iron in drinking water is good for your health but for a certain amount.presence of higher amount of iron can cause bitter taste and stain the laundry.
hope i answered your question well.

I agree with Mr.
Jida Jaanini about his answer above but I will add some important points to monitor water quality, we need to measure not only physical properties but as well as microbiological test mainly total coliforms and E.
coli with some chemical properties to be checked such as pH, Total Hardness, Ions mainly Calcium, Magnesium, Fluoride to be included on this test.

The multi-parameter water quality meter would be appropriate for use.
There are some of these applications that measures up to11 water quality parameters at the same time.
An example of this is the U-52 Horiba Water Quality meter.
Hanna9828 measures up to13 critical water quality parameters (6 measured and7 calculated.) Another good product is the Professional Water Quality Test Kit manufactured by PRO-LAB.
This kit allows you to perform up to sixteen (16) water quality tests.