Mining is important in that it is the only way to gain specific materials and in larger quantities. Many metallic looted items can be melted down or scrapped into their base materials but the output isn't much.
There are a few caves already available on the Random Gen maps that players can mine in but mining can take place pretty much anywhere by digging a hole in the ground. Top layers generally consist of Dirt, Fertile Dirt, Clay and Crushed Sand but lower levels will contain various amounts of Small Stone (primarily), Iron (often mixed with Small Stone), Lead, Lump of Coal and Nitrate Powder with occasional pockets of Crushed Sand. Oil Shade can only be found in the Desert biomes and is used to create Gas Cans.
Premade caves (Random Gen maps) will often contain stalagmites and stalactites that can be mined. They will product Potassium Nitrate. There are tools specifically meant for mining in the game that include Auger, Steel Pickaxe and Iron Pickaxe but other Axes and Tools can be used. Even Weapons and fists can be used, but the damage done to the ore may be minimal and it could take a long time.
An important thing to be aware of is where the digging and mining is taking place. If it is close to the surface and under buildings, the building and the ground in-between could collapse and kill you. It would be better to dig straight down at least 30 blocks before mining tunnels, and even then could be an issue depending on the locale. The ground above could still collapse. The lowest point in the game is -57 meters. This is covered in Bedrock that cannot be mined through. When a player hits Bedrock, the ore when hit will make a specific 'twang' sound. Structural Integrity is calculated from Bedrock and any activity that happens above it. Players will want to use support pillars in any large open areas to reduce the chance of collapse.
During the daytime, tunnels are fairly well lit but during the evening and night hours, they are completely dark and players will need to provide lighting in the form of Torches, Spotlights or by using a Mining Helmet. How the daylight from the sun reaches the mines and tunnels is unknown.
While zombies can't dig straight down (yet), they can spawn in ready-made caves. Mining also causes activity, which will spawn in a Screamer. The closer a player is to the surface, the more detectable the activity is and the higher the chance of attracting unwanted attention. Use a hatch to close off any tunnels leading straight up to keep zombies from falling in, although if it's deep enough, it will probably kill them anyway, but why take that chance.