Once you've researched Mining, possible mining sites appear on the map.  To use them, you seem to need to build a city next to them, much like building farms.
Also, it appears that with some tech, you get a bonus immediately when you start researching it (say 20%), and then every turn you research it, the bonus improves up to some maximum.  This is instead of waiting for 20 turns then suddenly getting a massiv bonus in one hit.  I like it this way.