The minimum kit
A copy stand or sturdy tripod with a center column. A digital camera with a macro lens (anything from a smartphone with a macro accessory to a full-frame DSLR works; the technique matters more than the gear). Two soft-box lights or LED panels you can position freely. A graduated or solid-color background — black for fluorescent specimens, dark grey for crystalline whites, off-white for dark sulfides. Compressed air for dust.
Lighting
Place lights at roughly 45° on either side of the camera axis, diffused. Avoid harsh shadows from a single bare light. For dark or metallic specimens, increase contrast with a small fill card opposite each light. For transparent crystals (fluorite, quartz), add a backlight to bring out the color from inside. Each species has a sweet spot you'll find with experiment.
Composition and post-processing
Shoot from the angle that best shows the crystal terminations, color zoning, and habit — usually slightly above the horizontal. Use focus stacking (multiple shots at different focus distances merged in post) for sharp detail across the depth of a 3-D specimen. Light post-processing only: white balance, mild contrast, dust spotting. Heavy color or saturation editing destroys the specimen's authenticity.