The Prospector's pick is a tool used only for locating ores, it cannot be used for mining. To make the prospector's pick you first need to make the tool head for it, and it can only be made out of metal. Either use a ceramic tool mold and pour smelted metal to make the head, or smith a metal ingot into a tool head at an anvil. The prospector's pick tool head can then be combined with a stick or a bone to produce a functional tool.
To prospect for ores, right-click on stone blocks. When you right click, it shows in chat message will show how many ores are in a 25x25x25 area centered on the block you right-clicked (ie. 12 blocks outward from the block you right-clicked). Be warned, the Prospector's pick returns false negatives frequently, so you'll want to check several blocks at a time. The levels from least to greatest are: nothing of interest, traces of (x), small sample of (x), medium sample of (x), large sample of (x), very large sample of (x), where (x) is the name of the ore in range. To maximize use of the prospector's pick, find a reading of ore, mark that spot and dig 12 deep tunnels in all 4 directions and mark the ends of those. Mark for the locations where readings transition to the highest concentration of ore, and find the center of that area. By pressing F3 you can get the exact xyz coordinates, and calculate where the ore should be located.
A detailed guide for prospecting, made by Kittychanley, can be found on this youtube link.
An older youtube guide can be found here (*Note: the blueprint for the tool head is no longer relevant in the current version).