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As for looking for abandoned mines, go online and just google your local area with terms like "gold rush" silver mine, gold mine, copper mine etc.
You can find info on mines across the us and canada on government websites, for canada its Minfile and for the states they each have their own websites, but USGS has info.
Next once you know there are old mines in your area, start looking around on google earth. go into the mountains or local area and start looking for old roads, follow them, look for collapsed buildings, switch backs and ore dumps, once you know what your looking for you start seeing them everywhere in some areas.
Next, legally mining claims, both placer or hard rock are still public land, basically if no one is there go ahead and explore, do not be a shit head and break into anything locked. Do not disturb or try and enter anything obviously active.