tinyE: That reminds me, is there a trick to mining? Sometimes I feel like I'm just aimlessly digging and sometimes I do, going for an hour without finding anything. I usually just get frustrated and start a new world.
I usually just explore caves (the ones on the surface often go down quite far; some even to Hell) and use something that can throw light past the walls to find ores and other caves, but you could always craft Spelunker potions that light up all the ores and chests around you (really useful for finding underground cabins) if that's what you're struggling to find, or the Hunter potion to light up mobs to find other caves. If you feel prepared for it, the underground jungle is fun to explore. I'd definitely recommend a Spelunker potion or 20 to help find the... hidden fun stuff (uh, not in the Dwarf Fortress sense, but actual fun stuff).
Crafting a bottle and placing it on a shelf, table, or workbench will allow you to craft potions near it. I think all of the buff type of potions require a bottle of water (crafted by having bottles while standing near water; basically, the water is your crafting station, and the bottle is the only ingredient), so show that to the guide to see how to make those potions.
Jonesy89's suggestion of a Hellevator is good too, though I'd make it 3 wide so you can free fall down one side and climb up with platforms on the other (if needed). As a bonus, it'll be useful later in the game when you're ready to tackle Hell. Bear in mind that Hell evaporates water, so stop before the background changes if you're using a water landing pad.
Edit: Oh, right, Jonesy89 mentioned rope as well. I guess 2 wide works with that while still allowing a free fall.