How does one upgrade their wand, and when is each upgrade available? I'm a bit of a dummy, I don't know if it's automatic over time or if after a certain milestone i have to press a button. Just asking in case I missed something.

The wand is upgraded as you "kill" ghosts, either by siphoning or eating them. You gain upgrades at 600, 1800, 4800, and 10000 kills, respectively. If you have the Delicious Perfume, ghost kills are doubled. If you have the Rich Siphon, there's a chance "kills" won't count towards your kill count but provide a bit of money instead. In my experience, you will likely get the 4800, Lvl 3 Wand, by the time you reach the final floor.