For this achievement, you will first have to upgrade your farmhouse one time so it has a kitchen. The farmhouse you are given at the start does not have a kitchen. This can be done by having 10,000 g and 450 wood. Once you have the gold and wood required, talk to Robin at the Carpenter's Shop north of Pelican Town. The upgrade adds a kitchen and a bedroom.
Once the farmhouse is upgraded, you will have to learn recipes to cook. The best way to learn recipes is to watch "The Queen of Sauce" on TV at your house. According to the Wiki, episodes begin on Spring 7 of Year 1 and a new recipe airs each Sunday for the first two years of the game. There are 32 recipes you can learn from the show. There are 74 total dishes in the game.
You will know if you already learned the recipe because the television menu will show "re-run" next to the show title. Also, Pelican Town residents will occasionally send recipes themselves. Another outlet for learning recipes is Gus, the saloon owner. He sells recipes you can learn. After that, the best way is to cook dishes where the items are common in the game. Bread, fried egg, omelet, pancakes, sashimi (any raw fish) are examples of easier dishes to make.