To get this achievement, you must collect 13 different mushrooms: one of each specific, naturally-occurring, one-block tall (i.e., non-elder) mushroom type. Unlike most achievements, there is no master version of the forager achievement.
The mushrooms must be naturally-occurring in order to count towards progress of achievement completion. In other words, mining a mushroom that has been placed (by you or someone else) will not count towards this achievement.
Be wary of players selling "natural mushroom finds" in exchange for rare or valuable items, as there is no way to determine whether the person placed the mushroom or found it untouched.
Note that only the regular (non-elder) version of a mushroom counts towards this achievement.