These are the four kinds of cultivable crops that currently (October 2015) exist:
Most crops can be found randomly scouring the lands, such as Crisphead Lettuce and Turnips in most Forests or Swamplands as well as Horned Melons on jungle floors. Wheat Seeds can be obtained randomly by collecting Tallgrass or Savannah Tallgrass (even after placing it yourself).
Seeds can only be planted on "tilled land" - you have to use the Plow on either Grass, Dirt or Mud to get tilled land. All seeds will also need Water, Bog Water or Mineral Water 1 block close to them to grow. The water can be placed underneath the tilled land as well.
Seeds and seedlings can be fertilized with Pigsy Droppings to grow faster, but they will stay fallow in any case in too cold (snowy, icy... wherever water turns to ice), too high (above ca. 150) or too hot (lava layer) environments as well as in Swamplands (not being the fault of the Bog water though, as normal water nearby also doesn't help).
Then you'll just have to wait until the seeds will grow into seedlings, sprouts and plants - only when ripe or grown you can harvest them.
"The design for crops is that they'll progress through a few states before you can harvest them, the last two states are Ripe, and then Grown. Ripe giving you the best return, and grown giving you less of a return but still giving you the final product (unlike harvesting a sprout, which only gives you seed). You have a little over 24 hours after a crop reaches the "Ripe" state to harvest it before it becomes "Grown". It's essentially the same thing as the crop withering, but with less of a punishment." - Thor 10-23-2015 See forum discussions