Children will begin writing by holding their crayon in their fist. As they begin to develop fine motor skills the grip changes into more of a pincers grasp. As the child develops the grasp changes and eventually becomes the more typical pincer grasp we are familiar with.
Young children draw straight and curvy lines to represent 'words'. This is encouraged. The teacher or parent volunteers will sit with a child and have them tell what they wrote. The adult writes exactly what the child says, in their own words. This creates the understanding that my print has meaning. Reading Rockets has a detailed article on the process of learning to write as-well-as helpful examples.