You usually have one at the start of every game. Train more in the University, using expert workers; they cost only $500 and the obligatory two pieces of paper, and they revert to being expert workers if disbanded.
Prospecting takes one turn. A barren area is marked with a pick-axe and red cross near the bottom whenever a prospector or miner or developer is looking around. A mineral has its icon displayed near the top if there is only one but one above the other if there are two. Only you can see minerals you have uncovered. After a miner develops them, there is an appropriately coloured triangular icon, which every player can see.
If you see no prospectable tiles on the visible section of the map, click an icon near the bottom of the right-hand panel and your new view will centre on a suitable tile if there are any left.
You may prospect anywhere in your empire and on any tile in a minor nation where you have an embassy. Discoveries in the minor nation can later result in land purchase by your Developer and extraction royalties thereafter (as with other types of tiles the Developer may purchase) or at least while the relevant province remains part of an independent minor nation.