No, you only have to do npm init when you're first creating a project. It essentially just creates the package.json file (https://docs.npmjs.com/cli/init.html).
And you should only have to run npm install when you first set up a project for local development, or when changes are made to the project's dependencies. So, usually just once, unless you've made changes. (https://docs.npmjs.com/cli/install.html)
npm start is a script that should be defined in your package.json, and you will likely need to run that every time you begin local development on your project.