Minecraft wiki entry for Things_not_to_do, associated text: Sticks: Fletchers buy 32 sticks for an emerald. Therefore, obtaining an emerald is as simple as mining 4 logs and converting them to sticks.
Carrots and Potatoes: Farmers give you emeralds if you give them crops. If you break carrots or potatoes with a fortune tool, you get more crops. This doesn't apply to wheat or beetroot. Carrots and Potatoes are a cheap way to get emeralds. Sweet berries are also available to sell, but only to Master level Butchers. Once available, they are a good way to get cheap emeralds.
Pumpkins: Pumpkins can be easily farmed fully automatically allowing the player to obtain emeralds easily once a simple farm is set up. Unlike melons or paper they don't even require crafting.
Mob drops: Some villagers buy mob drops for emeralds. It is easy to make a mob farm, making these items plentiful.
This includes rotten flesh dropped from zombies, as the player always gets hunger from eating it, it's good to sell it to cleric villagers.
Any raw meat, such as beef, porkchop, chicken, mutton, or rabbit can be sold to butcher villagers for emeralds. This is not a big problem if the player has a large animal farm, but it's important not to "cook" the meat by using fire aspect enchanted tools/weapons on the animals.
Any raw fish, such as cod and salmon, could be sold to fisherman villagers for emeralds. Like the mob farm, try not to kill any fish with fire aspect, a good and simple way to prevent that is fishing with a fishing rod.
Glass Panes: Cartographers buy 11 glass panes for an emerald. Glass is easy to obtain, as sand is easily found in a desert, or it can be obtained via wandering traders. However, it is sometimes hard to find wandering traders
Stone: Masons buy stone for emeralds. With a silk touch pickaxe emeralds can be obtained easily.
Wool: Shepherds buy wool for emeralds, in which the player can obtain easily if they have a large sheep/wool farm with different colored sheep for different colored wool.
Flowers: Shepherds also buy dyes from the player. Most dyes can be easily obtained through flowers, but sometimes the dye isn't easy to get, such as brown dye.