Okay shorter post here, mainly because mining is really in its infancy at the moment. The concept is simple: You need a Finder, some mining probes, and an extractor. Currently there is just one type of each so no concept of better equipment yielding better; rarer; or deeper finds. But, the mechanics are there and I'm sure in time this will be an area that is expanded upon.
Using your finder will drop an explosive probe an bring up a new radar screen. Should something be found it will be shown on this radar. Then it is just a case of heading over to it and using your extractor to draw out the resource.
The only thing to bear in mind is that the mining radar is fixed, so that north is always at the top of the screen (unlike the main radar, which rotates along with your character). So you just need to align yourself correctly before heading off to dig up your claim.