How do I build a home-made apparatus for identifying what minerals are present in water?
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)