Starting in Alpha 0.0.8, you now have access to exceedingly profitable mining activities! This guide will tell you everything I know and how to get started. :3 You can do this at any time while your shop is closed, so when you get the "You feel tired" message telling you the customers are gone for the night, you can go do this while you wait for morning instead of sleeping, if you like.
Update: 0.0.82 changed how time works, so unfortunately you will have to sleep. :( A lot of other things have changed too, so please stand by while I figure out what to update here...
Meet your pickaxe
If you want to mine anything at all, first you're gonna need a pickaxe. If you're new to My Little Blacksmith Shop, a copper pickaxe head has already been provided to you on the crafting table, otherwise start by buying 3 ingots of the highest tier you can afford, as well as any 2 handed grip. Heat the ingots and place them on the anvil, then select Axe Head and hammer away. First you will get a double sided battle axe head, which immediately tries to fall off the anvil into the oil. Quickly catch it and set it back on the anvil and hammer it one more time to turn it into a hefty pickaxe head. Then take it to the crafting table and combine it with your 2h grip. Finished! This pickaxe will be your new best friend from now on.
To wield it, you'll need to press E to add it to your equipment, then you can switch to it with the number 3. Whenever you can afford a new type of ingot, make yourself a new pickaxe with it. You can only mine ores that are of the same variety as your pickaxe or below.
To the mines!
Now you are ready to go harvesting rock. In the yard behind your shop, you should notice a new horse and carriage. This will be your main ore transport. If you want, you can optionally load an open-topped crate or two onto your wagon before we get going, which can help you transport ores out of the mine easier by letting you carry more than one at a time. If you don't have any crates, see the Shopping trip section below for where to find them -- They're free shopping baskets.
Give your horse a Left Click to set it to follow you, then head straight out through the forest. The forest is heavily fogged over, but the path is fairly straight, just follow the widest opening. Your horse is as slow as the default walking speed (I do not believe it upgrades with your agility), so when you get to the cave entrance, stop and wait for the horse to catch up. Don't worry, your horse can path-find its way around obstacles easily enough, and if the cart gets turned over it will right itself by going straight for a while.
Once you and your horse are at the cave entrance, carefully guide your horse through a U-turn so it's facing towards home. This is important, because the horse isn't very good at making tight turns and may topple the cart while doing so, so you want to get turned around before you start loading it up with precious ores just to be safe. When you're done, click your horse again so it waits patiently outside for you.
Go inside and explore! There's a little bend in the tunnel and then you'll find a room filled with minable ore rocks. You can harvest any ores that are the same type as your pickaxe or below and they will respawn tomorrow. I have no idea what tier "Coal Ore" counts as, so it may be minable by any pickaxe. The rest directly correspond to ingot varieties you're already familiar with.
Ores that were harvested will automatically respawn the next day as a completely random ore type. There is a chance that it may respawn as ores that are too high-tier for you to harvest -- if this happens, the ore will remain there unharvestable until you either upgrade your pickaxe or quit and reload the game. Any ores you can't harvest won't allow you to swing your pickaxe at them at all.
When you hit an ore, either a Stone or an ore chunk will pop out. I don't know of any use for stone, so you'll only want to take the ore chunks for now. The mining room is actually on a completely separate map from the rest of the game (Thank you, seamless Unreal Engine level streaming!) so don't leave any precious metals unattended here or they might despawn, even if you only leave the cave for a moment. Instead, you can gather up all your individual ore chunks in the bendy hallway first (anywhere past the wooden doorway but before the stairs should be fine) and then take them out to your horse and load them up in the wagon.
If you brought any open-topped crates with you as suggested, instead of moving the ores one at a time through the map transition, you can bring an empty crate down into the mine with you, put your ores into it, and then lift with your knees (by crouching) to carry it. Don't overfill it, as your ores will have to deal with physics bouncing as you carry it around -- about 10 ores is the most you should be able to fit in it without problems. As long as you can manage holding the crate without making the ores go flying out of it again, you can carry the entire crate and its contents with you back to your wagon, drop it in, and take an empty crate back down to continue mining. It's okay to only take a single crate and simply unload it directly into your wagon, although I like to take two. As long as the ores are in the wagon, they should be okay to travel home with you, even if they're not in a crate. Caution: Do not attempt to turn crates over when they're in close proximity to your wagon, as their rotation point is on the bottom and they're likely to knock your wagon over.
You'll also find a floating crystal of fear in the mining area. Since it doesn't obey gravity, you'll have to carry that one home with you by hand if you want to keep it. As far as I know, it still serves no purpose and cannot yet be used to infuse anything with magic or similar, so for now it's simply a pretty glowing thing you can hold to experience some interesting render effects. It will reset its position back to where you found it when you reload the game, however.
You'll also find what appears to be a door at the back of the mining area. As far as I know, this cannot be opened yet, and will likely lead to more dangerous monster-filled dungeons and deeper mines in the future. For now, all ores will spawn in this one area, so don't worry about the door.
Then you can just tell your horse to follow you again, and assuming you can still make out the way home, just walk it back with you. I personally find it by following the largest gaps between the trees. Make sure you go slowly and keep an eye on your horse, so you'll see if any ores fall out of the wagon along the way. Guide your horse along the side of your fence to park it sideways, if you like. (Careful not to let the horse rub directly up against the fence or your wagon will turn over!) You can take the horse inside your yard, but there's not a lot of room in there to turn it around later.
Grab your axe
Now that you've got a load of ores, you'll need to fuel the smelter. Unlike the forge, you power the smelter with wood and coal, so unless you happened to get lucky with coal ores spawning in your mine, you'll need an axe now. Make a one-handed axe with two ingots, the axe head type, and a 1h grip. I do not know if the material used matters at all, the most it's likely to affect is how much "damage" trees and logs will take before splitting for you. Equip it using the E key and it will go to your 2 key equipment slot.
Your shop came ready with 3 initial logs to start with, which should be plenty for your first week of mining trips. When you run out, it's time to fell more trees. (You may consider taking your horse for this!) Find a tree that's nice and brown, and begin chopping away at the base of the trunk. If it's harvestable, you should see a "100%" or similar indicator pop up as you hit it. Keep going until you hear a loud sound effect of the tree falling. Then, continue chopping away at the fallen tree until it runs out of health, which will split it into logs, ready for you to take home and use as firewood. (In the future, you'll likely be able to craft these logs into wooden shields and grips.)
You cannot run out of wood. If you somehow stay in-game long enough to fell all of the trees and burn all of the firewood (such dedication!), you can simply save and reload the game and your 3 initial logs and trees will all return.
Smelting time
Split a log into firewood by hitting it with your axe, then load a few pieces of firewood (or coal) into the glowing hole in the side of the nearby smelter. Put your ores in the upper compartment and turn it on with the red button on the side to melt all the ores down at once. Remember to turn it off again when you're done so you don't waste fuel!
It does not matter if you mix different ore types when smelting them, the ingots will all come out just fine. Set the ingots on the ground as they spawn so there's room for the next one to come out, they'll cool on their own just from sitting out, so don't bother giving them an oil bath or anything. Now you've basically got a complete shipment of free ingots, which you can load onto the shelves next to your forge and use them take the next few orders you get from customers. Since you didn't pay anything for these ingots, their entire cost is added to your profits, which will send your gold supply soaring, making it easier and easier to get to the next tier of pickaxe, to harvest more ores, to sell to more customers.
Shopping trip
If you really want to absolutely maximize your profits while minimizing your expenditures, a new NPC shop was added to town in this update. Heading out from your shop's main entrance, up the path and off to the right a little, is a mostly unlabelled building with a door you can open. Step inside and you'll find every kind of grip for sale, including new Common tier ones, which are incredibly inexpensive. (A common one-handed grip is only worth 1 gold!) I believe in the future you may be able to craft wooden grips yourself with woodworking, but for now you can buy them here and take them back to your shop to use. To buy anything, simply exit the shop with it. The shop will despawn the moment you step out of the doorway, taking whatever money you owe it, and then when you step back inside the entire shop's contents will return ready to be bought again.
There is an open-topped wooden crate just in the corner of the shop near the doors you can use as a bucket to carry more than one thing home with you at once, if you're careful. When you go to lift the crate, remember to lift with your knees: Make sure you are crouched before you grab it to minimize the chances that the contents will immediately go flying from a physics jump, then uncrouch to hold it level in front of you as you walk. This crate will also respawn with the shop, but it's completely free to take, for your convenience.
No, you can't "steal" from the shop this way by simply holding the items indirectly -- the shop seems to charge you when it despawns based on what items were left inside it able to be removed, rather than checking what you took outside. Unfortunately you can't return (or "sell") things to the shop either, so be sure to bring your shopping basket back with you on future shopping trips unless you want to end up with more of them.