Great game, I love it. Keeps me entertained for hours on end.
The game is built in a way that you can stay in your little world minding your own business and "not wake" anything. You can keep it simple or wander a little. If you wander a lot of course, you will be getting some monster's attention. Before you get too busy exploring, do yourself a favor and gather lots of stone and wood. Make stone walls and wood walls for your home by using a "Work Bench". You will soon notice that your little safe haven will be "invaded" by some friendly visitors who want to claim your home as their own. In short, move a bit away from your home and build a building with rooms for people to move into your world. You can assign them to their own homes (you should build at least 6 different rooms with tables and chairs). Your "Guide" will tell you, you need to prove yourself by defeating a strong monster.
You don't have the right axe to mine Ebonstone but, you can bomb it. Walking on the surface (left or right of your home) long enough will reveal the corrupt area (purple blocks with chasm)... DO NOT drop down in the chasm, instead you can bomb the entrance and patiently lay blocks to slowly descend. Bring plenty of torches. Your adventure begins....