Cooking in Harvest Moon DS is very similar to cooking in Magical Melody (GC). You basically have one cooking utensil to use to combine with various ingredients to make the cooked recipe.
To set up your cooking environment you'll first need to upgrade your house to the second stage. Once you have your house remodeled you can watch Channel 2 on your Sprite Station until you can buy the kitchen and it's matching utensils.
Once you've bought your kitchen you can begin cooking. There are also 5 utensils you can purchase from Channel 2. Press A when standing next to the kitchen to start cooking. You'll first select up to 8 ingredients (keep pressing Down through your rucksack menu to select ingredients stored in your refridgerator), then select the one utensil you want to use. That's all you have to do.
You can also ship the cooked food, but only if you put it in your Basket first. Then you have to dump the contents of your Basket into the Shipping Bin. You can also Can the food if you have the Maker Barn and the Cannery. Shipping your cooked food is required to complete your shipped item list, one of the requirements for the Harvest Goddess marriage.