Cultivator
The first thing you are going to need to garden is a cultivator. This can be found in the forge. In order to make this you will need 5 Core wood and 5 Bronze plates.
Cultivating allows you to till the ground and plant various seeds or you can use the cultivator to add grass to the area.
Seeds
Where you plant your seeds matter. Seeds need space to grow (think a couple in game feet between the seeds.) Seeds also need access to light (constant light is not needed. As long as the soil is in direct sunlight at some point in the day, your seeds will be healthy.)
Seeds that are hidden from the sun or too close together will be shown as unhealthy.
Tip: Seeds cannot be dug up and replanted. Once you hit them, they are gone. However, if your unhealthy seeds are just too close to each other and they have sunlight, you can simply wait to harvest the seeds around them and the unhealthy seeds should become healthy with adequate space.
Once your seeds are grown, you simply need to harvest them and start planting again.
To our knowledge plantable seeds are Carrots (found in black forest), Turnips (found in swamps), Flax and Barley (found in Plains and can only be planted in the plains biome).