Keep running left or right (on the surface, not below ground) until you find weird grey looking grass and unique flying enemies (eater of souls or something) and make your way down one of the many holes in the floor. They are usually very deep though so have a grapling hook before you try it tbh. Also you need to have dynamite with you (50s each from the demonlisionist) to destroy the ebonstone blocks in the way of you and the shadoworbs, but watch out as their blast radius is massive. When you destroy a shadow orb you usually get demonite ore and a blue grade item.
Main aim is to get the best gear you can (iron or gold armour + weapons) and then go to smash a few shadow orbs. If you summon the eater of worlds boss (giant multisegmented worm) and die in one of the pits, don't worry as you can continously fight it even in your house, unlike the eye of cthulu which despawns.
If you manage to kill the worm, you get demonite ore and a piece of the shadow armour + shadow scales. With the ore you can make demonite bars and with the scales + bars you can make weapons and armour, most notably the nightmare pickaxe, which allows you to mine the ebonstone which you currently need to use dynamite on.
Also when you destroy shadoworbs, you can have a meteor land somewhere near it which creates a hole nearby filled with ore that kinda looks like its on fire. Get as much of that as you can and you can start making some truly great items.
Thats a lot of hours though worth of gameplay there
Enjoy!