Mining Asteroids

you need to research tech level 10 in each resource tech to discover the remote mining components

Research Minerals Extraction, Organics extraction, or Radioactive extraction to TL10+ to get the remote miner component put that on the ship and station it in orbit of an uninhabited planet or in an asteroid hex and you will start mining. You will incidentally also have a good boost to planet based resource extraction going from TL1 to TL10

I believe I read somwhere its more efficient to build mining satellites rather than mining ships. Something to do with maintanance. I have yet to get to this level myself. I am playing slowly waiting for more bug fixes.

These require less Maintenance than a ship but obviously are smaller (will fit only one miner unit) and will require you to build a ship to deploy them from they are also more difficult to redeploy. You can of course build the satellites in any colony which has the cargo space.

So its not that straight forward

Or do it the easy way, build a ship with a yard a sat bay and a few cargo bays, then just set it making and launching, tis easy. (would be even easier if you could get a minister to do the building / launching for you, maybe you can I just never use ministers for anything).