Start with water
Warm water, a soft brush and patience remove most dirt. Do this first and reassess before reaching for anything stronger.
Check solubility before any water contact at all. Halite and several sulfates will simply dissolve.
Acids are a last resort
Oxalic acid removes iron staining from quartz and is the usual first chemical step. It is also toxic, and it destroys carbonates, so any calcite in the specimen is gone.
Work outdoors or under extraction, wear protection, and neutralise and dispose of the waste properly. Never mix cleaning chemicals.
Mechanical methods
Ultrasonic cleaners shatter fragile and cleaved specimens; they are for durable pieces only. Air abrasives are effective and unforgiving in unpractised hands.
Test any method on a specimen you can afford to lose. There will be one.
Decide whether to clean at all
Not every specimen is improved by cleaning. Iron staining is sometimes original to the occurrence and part of what makes a piece recognisable from its locality, and removing it can reduce both interest and value.
Old specimens with historical labels are the clearest case for leaving alone. Cleaning is irreversible, and the version you are looking at has already survived a century in that condition.