Possibly one of the best combinations of professions in the game, in terms of making money. Unlike a lot of professions, you will be spending very little money on components for recipes, and be able to gather your own resources. The other profession involved alongside these two is the Tailor profession, but this is only useful if you have the Woodcutting profession aswell (or Forester in basic terms), so for now, stick to the metal, because you'll need to purchase components from other players for the Tailor profession. Either that, or make an alternative character with the Forester profession.
That out of the way, lets get to the nitty-gritty of Prospecting first of all. This is an essential skill to train if you're going to make it as a Metalsmith. Your first ores will be Copper and Tin. I found it best to begin life as a Hobbit, as Hobbiton is riddled with Copper and Tin deposits. So you'll notice a new skill you have on your skills bar called 'Track Mines'. Press this once, and the skill will be activated. It's incredibly straight foward to utilise. Simply run about the landscape (preferably Hobbiton or East of Bree- near the Midgewater Marshes). Keep an eye on your mini-map. Eventually you'll notice a dark blue arrow appear somewhere on the circumference of the round mini-map. Go towards this arrow, and eventually you'll see a pickaxe crossed with a smithing hammer. Hold your mouse over the icon, and the type of deposit that you have found will appear on screen. Simply move to the icon, and you'll see the deposit on screen ready to mine. Equip your Prospector Tools and right click the deposit. Your character will begin to mine the deposit.
Once the Prospecting bar is full, a box will appear indicating what can be found in the deposit. It's easiest just to click 'Loot All', as the likes of Pale Brimstone can be sold to vendors, and later on, will become useful in the Metalsmith Profession. The frequency of deposits you find will vary. For instance, it is far more difficult to find Tin Deposits as oppose to Copper Deposits. It doesn't get anymore straightforward than that.
But what do you do with these ores? This is when you need to find a furnace. You can begin smelting the ores, which means taking out the impurities, and leaving pure metal behind. You'll start by smelting Copper and Tin ingots. You will require two ores of each metal to make one ingot of that metal.
2 Copper ores = 1 Copper ingot
2 Tin ores = 1 Tin ingot
Once we have our ingots we can make armour with them. But Tin and Copper armour? Surely not. No, we have to smelt 2 Copper ingots, and 1 Tin ingot to make two Bronze ingots. This is the only instance where you will recieve two ingots as a final result. Other ores, such as Barrow-Iron, will only produce one ingot.
Once you have these ingots, you can begin to make armour, in this case, Bronze armour. Quite simply, you have to retrieve the required ingots to make the selected item. Make sure you have your smithing hammer equipped!
That's pretty much basic Prospecting and Metalsmithing. The same rules apply to higher tiers of each profession respectively. Once you have become proficient at the current tier, you will unlock the next tier of recipes. If desired, you can continue to make ingots or armour on the tier and earn mastery, which will allow you to produce more ingots from ores, or better armour from ingots. You can increase chance of this happening by using items such as Brimstone in the recipe.
As you unlock each tier, there are a few tasks to complete to be bestowed with the skill the next tier requires. This usually requires visiting Glasi Ironhand in Thorin's Hall, which is easy if you're a dwarf. Further accquitances will be Ottar and Jaki Stonehand as you reach the higher tiers.