You have to have a corn in your inventory or workbench to plant it. If you don't you'll get the message 'needs dirt'. Each fruit or vegetable you find will plant one plant that will regrow when you harvest it. For carrots you need a Carrot, not a Carrot Flower which you will also find.
You have to assign a settler to your crops but they will tend to all the plants of that particular crop, so you don't need a settler per individual corn plant, they'll tend the entire patch of corns.