CMDR Thrust from Bradford's guide to mining
Mining is really cool, even though I probably made many rookie mistakes to be one of the first to try it

!
I went to a ring around one of the stars, these are now selectable. Getting there in super cruise got pretty toasty, so I worked on my tan as well (who says guys can't multitask), however once out of supercruise the temperature dropped.
I bought a mining laser, and stuck it on one of my small mounts on the cobra. I also bought a refinery with three bins (more on that later) and a fuel scoop and a detailed surface scanner; this only left me 8T of cargo space since I didn't find the 16T rack.
Blasting an asteroid in the same area for a while breaks off little chunks of rock... This can be tricky because they spin, meaning you should find a spot with less spinning. Maybe having multiple lasers helps if they are well grouped, but the effective range is pretty short. Hitting multiple spots at once could also work, having said that!
You can scoop up one chunk at a time with the cargo scoop, and targeting it gives you an idea of its composition a long with yield of most common mineral/metal.
Once inside, the refinery separates tells you what's in there, and you have the choice of sorting it into one of your bins, or venting it if you don't have room. Each bin can hold one resource (bertrandite, gallite etc.), and scooping aster-chunks with those resources fills up a progress bar, giving you 1T of that resource when it's full. These chunks seem to be 1T blocks, because if it is 15% bertrandite and you are collecting it! the progress bar goes up by 15%.
Each asteroid seems to give different resources, so it's a good idea to pick one and go for it unless you have lots of empty bins, or like wasting resources by venting stuff you don't have room for
My first attempt just messing around took me about 10 minutes of blasting until I got 1T bertrandite and 1T of coltan. I sold these for about 4000 credits. Not bad, considering this was in a well known system that is probably mined out

!
Oh yeah the surface scanner didn't do anything to the asteroids, but maybe scanning the planet could give clues about what's in the rings?