heh, first research Construction, then research Mines
Then build a ship with some cargo bays and a minelayer component. Then design a mine, and build some. I suggest you research Explosive Warhead 2, to get more "bang for your buck" (hahahaha).
Move the minelayer to the planet where you built the mines, then press the "T" key to transfer cargo. Transfer the mines from the planet to the minelayer ship.
Move the minelayer ship to the sector you wish to mine, and press the "U" key to see the Deploy/Retrieve Units screen. Click on the mines to deploy them.