Now that you've got your home, you can begin collecting materials. Your inventory should look like this for both the low and high risk options.
LOW(ER) RISK OPTION:
Make a separate room in your home, sealed off by at least one door. Start burrowing down.
Luckily, you'll run into a cave or some veins along with way. If you don't, make some random sideways tunnels branching out from your stairway. In this case, I hit a cave.
HIGH RISK OPTION:
Charge headlong into the nearest cave you can find. Start placing torches along the sides until there's no darkness left. Your cave should now look like this:
OH BOY! Iron!
Once you come across iron, bring it back up to your base and smelt it down. You'll need this to craft iron tools, of which the pickaxe can mine gold, redstone, and diamond ore.
Congratulations, you now know how to mine! Watch out for monsters!