Ziggurat

The wand is the default weapon; the one that uses purple mana and recharges without the need to collect crystals. Apprentice is the starting character, so don't pick any unlockable characters. Be sure to select Easy on the character select screen as well.

This Achievement unlocks once the third floor starts (not at the end of the second floor), assuming you didn't use any other weapons by mistake. To make it easier to avoid using other weapons, just don't pick any up! The Achievement is fairly easy as long as you don't get an unfair room or boss during the first two floors.

I'm not sure whether amulets count against this Achievement, but I recommend not using them just in case.

To make this achievement easier, I'd recommend playing the game until you unlock the perk that sacrifices a weapon to double wand damage. Once you have that perk unlocked, you can collect whatever weapon spawns at the beginning of your run and hope that through level-ups and hidden rooms you gain the ability to sacrifice that collected weapon for double wand damage.

This might take several tries, so what I recommend is to unlock the perk, collect the random weapon that spawns at the beginning (but do not use it!), and just play the game on the easiest difficulty, using only your wand. Explore every room and kill every enemy for level-ups and be on the lookout for cracks in the walls to find hidden rooms that have perks in them. Eventually things will work out in your favor and you will earn the perk and double your wand damage, making this achievement a breeze. Worst case scenario,you never get the perk, but you pull off the achievement without the perk by being much better at this game than I am.