I am a regular substitute (10 yrs) for the elementary school my daughter attended. It is am exemplary school, Blue Ribbon school and well recegnized... people pay to transfer in to this school. We chose this area to live based on the school system.
I don't know about nationwide but starting in K, the children are taught the proper way to write the alphabet, names, etc.. As they move from K -1 the writing gets more detailed, they have journals, they have something to write daily in the journals and the teachers read it and comment on EACH entry.
Moving on to 2nd grade, a lot of the children are anxious to learn cursive and start on their own. There is another level of more writing in 2nd grade.
3rd grade is when the cursive is practiced daily and once the student gets through the practice cursive book and passes everything then they no longer print... ALL writing is in cursive. They have creative writing, journal writing, etc. State testing begins in 3rd grade and part of the test is writing.
This past week, the 4th graders were on writier's workshop and all the classrooms were turned into a "campsite", they wore special t-shirts, teacher dressed like camp counselors and the focus of the 2 days was the creative writing they will be tested on as well as cursive.
It goes on up the scale that way. My daughter is in 11th grade at a Sr high school which is recoginized in the top 1% of high schools in America. Yes, there are days she does take her MacBook to school for writing projects but on all major tests, they have "free response" which is writing.
Also, begining in K, the children work on a typing program "Paws in Typing Town" to learn typing as well. The computer is used a LOT but the students here also expected to have good handwriting skills.
Keep in mind, the parents are encouraged to work with the children and most in our area do. Parents need to work with the children with all aspects of their school work.