How do I build a home-made apparatus for identifying what minerals are present in water? While there are many readily available kits in the commercial market for testing the type of minerals present in water, 


is there a way we can build one at home? 


I am ready to spend some bucks on raw materials, if I can get the knowledge of how to build such apparatus
This probably not what you are looking for, but the cheapest and least time-consuming analysis is the one done by others for free.

Over here (Germany), local water suppliers are required by law (Trinkwasserverordnung - TrinkwV) to monitor the quality of potable water. This includes a large number of microbiological and chemical parameters. The details are given in the annexes 1-5 of the TrinkwV.

The results are published and are are available online at no cost. 

Examples (mostly in German) are:


RheinEnergie (Cologne area)
RWW - Rheinisch-Westfälische Wasserwerksgesellschaft (one of the largest suppliers in Germany)