I am on Peaceful mode and using a Wooden Pickaxe and whenver I mine Iron Ore, it just disappears. Am I doing something wrong?
Wooden picks aren't strong enough to mine iron, you need at least a stone pick to do that. All you ever wanted to know about picks.
If you make a wood pick first and mine some stone then you can make a stone pick. A stone pick can be used to mine Iron and Lapis Lazuli ore.
Mine some iron ore with your stone pick. Smelt the ore in a furnace and use the iron ingots to make an iron pick. Iron picks can be used to to mine Diamonds, Gold and Redstone.
A Diamond pick can be used to mine Obsidian.
Wooden pickaxes aren't the best for iron ore. Instead you need to mine 3 stone blocks, craft them into a stone pickaxe in the crafting table, then you can mine the iron ore with the pickaxe. Depending on the type of pickaxe, it's best to use an improved pickaxe for the rare ores, such as an iron pickaxe, which is required by smelting the iron ore in a furnace into ingots. If you have 3 or more, craft it into an iron pickaxe to mine rare ores. Diamonds pickaxes are used to mine obsidian, but I will act like you're using your hand to punch stone, so it's best if they're enchanted.