Where you may legally collect
Updated
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The first question in this craft, and the one where confident wrong answers circulate freely.
The answer depends on who manages the ground
There is no single rule for "public land". Several agencies manage land in the United States and each has its own answer, sometimes on adjoining parcels.
So the question is never "is rockhounding legal" but "who manages this specific ground, and what do they allow".
BLM
The most permissive and the largest. BLM generally allows the public to collect reasonable amounts of rocks, mineral specimens and semiprecious gemstones for noncommercial purposes.
Reasonable is defined: 25 pounds per day plus one piece, and not more than 250 pounds in a year.
Excluded even on BLM ground: developed recreation sites, areas under active mining claim or other authorized mineral use, and land where the mineral estate is privately owned.
More than the reasonable amount requires a permit.
National Parks
No. 36 CFR 2.1 prohibits possessing, destroying or removing from a park any mineral, rock or paleontological specimen except as specifically authorized.
That is the whole answer and it does not have a personal-use exception. National Monuments and other Park Service units are generally the same.
Everything else
Forest Service land generally permits limited personal collecting, with rules that vary by forest.
State land has fifty answers.
Tribal land requires the tribe's permission and is not public land.
Private land requires the owner's permission, and a fee-dig site is private land with permission built in.
Wilderness, monuments and special designations restrict further.
The land manager for the ground you are standing on is the authority. Ask them, not a forum.
The commercial line
The one collectors most often miss. The BLM allowance is for noncommercial collection: selling what you collected under it requires a permit or a mining claim.
Which means a hobby that turns into a business changes the legality of the collecting that feeds it, retrospectively in effect, and that deserves settling before the first sale rather than after.