Where the good stone actually comes out of the ground, worldwide: turquoise, opal, jade, benitoite and more, with the fee-digs open to the public marked separately.
Many of the sites listed here are fee-dig operations open to the public, where you pay a day rate and keep what you find. Others are historic mines on private or protected ground that can only be looked at. Each page says which.
It depends on the land. Casual collection of small amounts is allowed on much BLM and National Forest ground; national parks are closed to it entirely, and claims are private property. The map shows land status and active claims.