Beryl can be found in alluvial deposits from erosion out of pegmatite veins. This is the easy way, sorting through sand and gravel. The hard way is to attack the granitic pegmatite with tools and see what's inside.
Depends on what the mineral is, and where it is located. Some are mined by conventional surface mining with power shovels, some are mined in underground mines, and some are mined by drilling a shaft,...