You may have already noticed that you're not in Kansas anymore, Dorothy.
Resources are not evenly peppered throughout the entire world, each kind is in a specific terrain type. Quite often you can sink a mine shaft and find nothing. Although caverns and abandoned mines do exist in many places.
You need to go find resources by exploring and looking for tell tale signs on the surface, or in the terrain. You may have to travel for a few in-game days to find what you want. One approach is to travel light, with a bed, dig a hole each night to skip night time.
The first question most players ask is, "Where be the resources?" And by that they usually mean ores. But animals and plants are also dependent on certain terrain types. From Dooglamoo's explanation:
You should be able to find some ore if just the required conditions are met, but you could find significantly more ore if the bonus conditions are met.
Required conditions: Stone base rock and thick topsoil layer.
Bonus conditions: Wet climate and bowl-like depressions in the ground.
Required conditions: Stone base rock and hilly or mountainous.
Bonus conditions: Smooth land with no visible erosion and with lava lakes or obsidian on the surface.
Required conditions: Granite base rock and deep eroded canyons.
Bonus conditions: Cold climate and hilly or mountainous.
Notes: Look for the stone strata layer in the granite base rock.
Required conditions: Granite base rock and hilly or mountainous.
Bonus conditions: Dry climate and very thin topsoil layer.
Required conditions: Granite base rock.
Note: Look for the andesite strata layer in the granite base rock.
Required conditions: Granite or diorite base rock.
Note: Look for the andesite strata layer. Emeralds are sparse.
Required conditions: Diorite base rock and smooth land that is far from any visible erosion.
Bonus conditions: Very thin topsoil layer and no lava lakes or surface obsidian.
Notes: Look for the granite strata layer of the diorite base rock.
Required conditions: Diorite base rock and lava lakes (possibly surface obsidian).
Notes: Diamonds are deeper underground and uniformly spread out.
Required conditions: Deep eroded canyons, no lava lakes, and thick topsoil layer.
Required conditions: Sandstone base rock and wet climate.
Bonus conditions: Bowl-like depressions in the ground (may be filled with water) and very thin topsoil layer.
Notes: Look for white terracotta strata layer in sandstone base rock.
Required conditions: Sandstone base rock, very dry climate, and very thick topsoil layer.
Bonus conditions: Deep ocean.
Found in the walls of caverns and abandoned mineshafts. You will need to tunnel around to find these as they mostly have no surface entries.
These can be found in the caverns that also contain trees and grass.
Resonating Ore and Resonating Crystals
These can be found in the walls of caverns and abandoned mineshafts. But are rare, so you may have to hunt for a long time to find some.