You need to craft "Something Delicious" at Lace's alchemy station in Shiptown. Then return with the Slime Snack to the gym in Safen.
You mean the turquoise shopkeeper slime you can buy the Slimophobe from? You can't do anything with him, I'm afraid. He is just there to foreshadow a later encounter in the game. Back when he was first added, his main purpose was to have someone to sell all the slime goo to you can loot from the slimes in the forest, since the goo didn't serve any other purpose back then.