Gems can be found by exploring the Dark Horse Mines.   Each time you explore here it uses 5% player energy and you have a chance to find: Nothing, Gems, Bits, or Ingots.  Players can increase the odds of finding Gems in the Mine by using a Lucky Rock from their storage room.   Lucky Rock's can be bought from the general store year round for 250 ingots or earned from completing every 3rd level of the Gems Found Achievement. 

Dyes can be found by exploring the Hoofbeat Plains (or other seasonal maps).   There are 2 different categories of dyes:  Base and Seasonal.  Seasonal dyes are further split into Spring, Summer, Fall, Winter and can be found in their respected season.  Exploring the plains requires the player to have 1 or more explore horses.   Each move on the map uses 2% horse energy, and actions like picking up items uses 5% player energy.