As far as mining goes, first you need a mining laser, this doesn't come with the worm by default that you start with so you need to buy one, or start with another ship.
Second ask for directions to the local asteroid field, it'll likely be one or two jumps away.
When you have the directions and your auto pilot is on, you'll hear your ship turning and be told when to jump do this with the p key.
When the asteroid field is ahead, fly into it and look for asteroids. If you have the radar (again the worm doesn't come with this by default), you'll hear near by objects as a honking sound in the sterrio field, if not,then turn around until you scan an asteroid then let fly with your laser by pressing the I key, center the targeting sound to fire accurately.
most asteroids will split into meteors when you hit them so you'll need to target the meteors and hit them again.
after you shoot a meteor you'll either have left behind some valuable cargo you can pickup like gold, cobalt or lithium, or a rock. if its a rock, just blast it, but if its cargo then fly to it and pick it up.
Nebula mining is even easier, just ask directions to the local nebula then simply fly around and pickup anything that isn't methane.
As to mission directions, I'm only just trying out the missions myself so I'm not sure how they work, I'm just doing the one with the missing scientist, though mostly it seems the majority of the game is concerned with gathering resources and either pirating or getting bounties so I've kept the missions until I was sure what I'm doing.
I'm now saving up enough cash for a gekko ship.
With our dreaming and singing, Ceaseless and sorrowless we! The glory about us clinging Of the glorious futures we see,
Our souls with high music ringing; O men! It must ever be
That we dwell in our dreaming and singing, A little apart from ye. (Arthur O'Shaughnessy 1873.)