The first procedure that can be done is to take a water sample and send it to one of the water laboratories in the relevant departments such as the groundwater department, the environment department, the laboratories of the faculties of science, agriculture, etc., in order to provide you with the physiochemical information of the well water.
Well, if you just want to measure the salinity in a sample of water, you can simply use a TDS meter, and so you'll get a measurement of the sample salinity.
however, if water quality is to be assessed, it is necessary to measure the major ions, the heavy heavy metals and the bacteriological analysis of the water, in order to judge the water quality and its suitability for different purposes.
A handheld TDS meter will solve the purpose. They are usually low priced (~ USD 3-5 , and readily available (on Amazon) and provide a good indication of the salt content.
You can get a refractometer as well to check the salinity (another low-cost approach), though TDS measurement provides a better sensitivity