The easiest way by far is to use the Prismatic Hammer
science weapon, with a science skill of at least 80
(thanks to everyone who provided additional feedback on this). Your basic attacks cycle through different types of elemental damage, while sweep attacks apply multiple types at once. The weapon description says the number of damage types the weapon can cycle through is determined by your science skill (mine was 80 when I got the achievement).
The Prismatic Hammer can be found on Groundbreaker really early in the game. Here's a video with the location:
Once you have the hammer just swing away at a mega or named enemy.
All credits for the video goes to RPG Division on youtube.