Mining for Ores
There are many ways to start mining once prepared, but there are three basic ways.
Find a cave that goes to the surface. Start exploring and work your way downwards. Caves offer a lot of already excavated stone, so more ores will be exposed. In caves, however, there can be many monsters. Water and lava in caves can also be troublesome.
Create a waterfall using a water bucket down a shaft or ravine. The waterfall will allow you to swim down and up. You can also build simple block stairs. Work downwards and explore, but there might be the hazard of falling monsters. Pillar jumping is a good idea to go up and search for tunnels.
Make your own entrance downwards. Dig your own way downwards from your base. Start digging horizontally at suitable depths. Be careful not to fall out of the world into the Void. This will take more tools, but with proper lighting, monsters will be rare. Your shafts might open into caves.
Try to mine everything you can find. Build a rail system or use torches, signs, or other items to mark your path.
It is a good idea to carve out a small space to keep furnaces, a crafting table, a bed, and a chest. If you plan on staying for a long time, make a small base with basic supplies. A bow and some armor along with a sword will protect you from monsters. Pay special attention to navigation, and keep a water bucket with you to extinguish any fire and form obsidian surfaces on lava. Dead-ends might be a thin wall that will reveal continuing passages. Build stairs to easily move around. Jack-o'-lanterns, torches, and redstone can mark unexplored tunnels, and fences and doors can block off water and lava.