High-school students are planning marches and school walkouts across the country in the coming weeks and months as the number of protesters on social media grows, galvanized by last week’s school shooting in Florida that left 17 dead.

Students at Marjory Stoneman Douglas High School in Parkland, Fla.—using various social-media platforms and hashtags such as #NeverAgain and #Enough—have encouraged students and antigun activists to organize. A nationwide walkout by teachers and students is planned for March 14, marches for March...