How can I make my tools last longer in Minecraft?

Enchanting!

The unbreaking enchantment will make your tools last longer by adding a chance that using it will not decrease its durability.

Repairing!

Placing two of the same tool in the crafting window will combine them into one item with 10% (of the original maximum) more uses! The two original tools must also be made of the same material.

Warning: Repairing enchanted items will cause any enchantments to vanish!

There are three methods of doing this.

1. Enchanting the tool.

The 'unbreaking' enchantment will make the occasional use of the tool not reduce durability, how effective this is is random, but I higher level enchantment will increase the chance of a 'free use'.

2. Repairing tools

If you craft two un-enchanted tools together you will get more durability out of the result. The equation for this is:

Tool 1 durability + Tool 2 durability + (Maximum durability/10) = Resulting durability

Be warned repairing an enchanted item removes the enchantment.

3. Anvils

Using a tool with an anvil can repair tools, either use two of the same tool, or a tool with its components (e.g. Iron axe and iron) be sure to try swapping the items round too, as it may be cheaper xp wise. This also has the added bonus of combining the enchantments.

Side note: Contrary to what is said in the question, using the wrong tool does not cost extra durability, only using them as weapons does.